direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C24.3C22, (C2×D4)⋊4C28, (C2×C28)⋊29D4, C2.9(D4×C28), (D4×C14)⋊16C4, (C2×C42)⋊5C14, C28⋊6(C22⋊C4), C14.110(C4×D4), C23.9(C2×C28), C24.6(C2×C14), (C22×D4).2C14, C22.39(D4×C14), C14.39(C4⋊1D4), (C23×C14).3C22, C14.136(C4⋊D4), C14.65(C4.4D4), C23.63(C22×C14), C22.39(C22×C28), (C22×C28).576C22, (C22×C14).454C23, (C2×C4×C28)⋊18C2, (C2×C4)⋊6(C7×D4), (C2×C4⋊C4)⋊4C14, C4⋊1(C7×C22⋊C4), (C14×C4⋊C4)⋊31C2, (D4×C2×C14).14C2, C2.5(C7×C4⋊D4), C2.2(C7×C4⋊1D4), (C2×C22⋊C4)⋊3C14, (C14×C22⋊C4)⋊7C2, (C2×C4).44(C2×C28), C2.8(C14×C22⋊C4), C2.3(C7×C4.4D4), (C2×C28).265(C2×C4), (C2×C14).606(C2×D4), C14.95(C2×C22⋊C4), C22.24(C7×C4○D4), (C22×C14).20(C2×C4), (C22×C4).90(C2×C14), (C2×C14).214(C4○D4), (C2×C14).226(C22×C4), SmallGroup(448,798)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C24.3C22
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=1, f2=e, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >
Subgroups: 466 in 258 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C24.3C22, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, C23×C14, C2×C4×C28, C14×C22⋊C4, C14×C4⋊C4, D4×C2×C14, C7×C24.3C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C4○D4, C28, C2×C14, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×C28, C7×D4, C22×C14, C24.3C22, C7×C22⋊C4, C22×C28, D4×C14, C7×C4○D4, C14×C22⋊C4, D4×C28, C7×C4⋊D4, C7×C4.4D4, C7×C4⋊1D4, C7×C24.3C22
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 133)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 176)(16 177)(17 178)(18 179)(19 180)(20 181)(21 182)(22 184)(23 185)(24 186)(25 187)(26 188)(27 189)(28 183)(29 160)(30 161)(31 155)(32 156)(33 157)(34 158)(35 159)(36 175)(37 169)(38 170)(39 171)(40 172)(41 173)(42 174)(43 199)(44 200)(45 201)(46 202)(47 203)(48 197)(49 198)(50 115)(51 116)(52 117)(53 118)(54 119)(55 113)(56 114)(57 111)(58 112)(59 106)(60 107)(61 108)(62 109)(63 110)(64 105)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 152)(72 153)(73 154)(74 148)(75 149)(76 150)(77 151)(78 143)(79 144)(80 145)(81 146)(82 147)(83 141)(84 142)(85 139)(86 140)(87 134)(88 135)(89 136)(90 137)(91 138)(92 123)(93 124)(94 125)(95 126)(96 120)(97 121)(98 122)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(204 222)(205 223)(206 224)(207 218)(208 219)(209 220)(210 221)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 217)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(36 222)(37 223)(38 224)(39 218)(40 219)(41 220)(42 221)(50 77)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(64 84)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(99 143)(100 144)(101 145)(102 146)(103 147)(104 141)(105 142)(106 133)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 148)(120 140)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(155 199)(156 200)(157 201)(158 202)(159 203)(160 197)(161 198)(162 189)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 204)(176 196)(177 190)(178 191)(179 192)(180 193)(181 194)(182 195)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(71 84)(72 78)(73 79)(74 80)(75 81)(76 82)(77 83)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 126)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(127 140)(128 134)(129 135)(130 136)(131 137)(132 138)(133 139)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(183 196)(184 190)(185 191)(186 192)(187 193)(188 194)(189 195)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 77)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 39)(9 40)(10 41)(11 42)(12 36)(13 37)(14 38)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 29)(22 220)(23 221)(24 222)(25 223)(26 224)(27 218)(28 219)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 96)(65 97)(66 98)(67 92)(68 93)(69 94)(70 95)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 120)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(127 152)(128 153)(129 154)(130 148)(131 149)(132 150)(133 151)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 177)(156 178)(157 179)(158 180)(159 181)(160 182)(161 176)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(183 208)(184 209)(185 210)(186 204)(187 205)(188 206)(189 207)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)
(1 39 77 8)(2 40 71 9)(3 41 72 10)(4 42 73 11)(5 36 74 12)(6 37 75 13)(7 38 76 14)(15 86 30 84)(16 87 31 78)(17 88 32 79)(18 89 33 80)(19 90 34 81)(20 91 35 82)(21 85 29 83)(22 61 220 52)(23 62 221 53)(24 63 222 54)(25 57 223 55)(26 58 224 56)(27 59 218 50)(28 60 219 51)(43 65 211 97)(44 66 212 98)(45 67 213 92)(46 68 214 93)(47 69 215 94)(48 70 216 95)(49 64 217 96)(99 177 121 155)(100 178 122 156)(101 179 123 157)(102 180 124 158)(103 181 125 159)(104 182 126 160)(105 176 120 161)(106 171 115 162)(107 172 116 163)(108 173 117 164)(109 174 118 165)(110 175 119 166)(111 169 113 167)(112 170 114 168)(127 208 152 183)(128 209 153 184)(129 210 154 185)(130 204 148 186)(131 205 149 187)(132 206 150 188)(133 207 151 189)(134 199 143 190)(135 200 144 191)(136 201 145 192)(137 202 146 193)(138 203 147 194)(139 197 141 195)(140 198 142 196)
(1 141 85 151)(2 142 86 152)(3 143 87 153)(4 144 88 154)(5 145 89 148)(6 146 90 149)(7 147 91 150)(8 207 21 197)(9 208 15 198)(10 209 16 199)(11 210 17 200)(12 204 18 201)(13 205 19 202)(14 206 20 203)(22 173 211 155)(23 174 212 156)(24 175 213 157)(25 169 214 158)(26 170 215 159)(27 171 216 160)(28 172 217 161)(29 195 39 189)(30 196 40 183)(31 190 41 184)(32 191 42 185)(33 192 36 186)(34 193 37 187)(35 194 38 188)(43 177 220 164)(44 178 221 165)(45 179 222 166)(46 180 223 167)(47 181 224 168)(48 182 218 162)(49 176 219 163)(50 126 70 106)(51 120 64 107)(52 121 65 108)(53 122 66 109)(54 123 67 110)(55 124 68 111)(56 125 69 112)(57 102 93 113)(58 103 94 114)(59 104 95 115)(60 105 96 116)(61 99 97 117)(62 100 98 118)(63 101 92 119)(71 140 84 127)(72 134 78 128)(73 135 79 129)(74 136 80 130)(75 137 81 131)(76 138 82 132)(77 139 83 133)
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,133)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,183)(29,160)(30,161)(31,155)(32,156)(33,157)(34,158)(35,159)(36,175)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,199)(44,200)(45,201)(46,202)(47,203)(48,197)(49,198)(50,115)(51,116)(52,117)(53,118)(54,119)(55,113)(56,114)(57,111)(58,112)(59,106)(60,107)(61,108)(62,109)(63,110)(64,105)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,152)(72,153)(73,154)(74,148)(75,149)(76,150)(77,151)(78,143)(79,144)(80,145)(81,146)(82,147)(83,141)(84,142)(85,139)(86,140)(87,134)(88,135)(89,136)(90,137)(91,138)(92,123)(93,124)(94,125)(95,126)(96,120)(97,121)(98,122)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(204,222)(205,223)(206,224)(207,218)(208,219)(209,220)(210,221), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(36,222)(37,223)(38,224)(39,218)(40,219)(41,220)(42,221)(50,77)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(64,84)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(99,143)(100,144)(101,145)(102,146)(103,147)(104,141)(105,142)(106,133)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,148)(120,140)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,189)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,204)(176,196)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(71,84)(72,78)(73,79)(74,80)(75,81)(76,82)(77,83)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(127,140)(128,134)(129,135)(130,136)(131,137)(132,138)(133,139)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(183,196)(184,190)(185,191)(186,192)(187,193)(188,194)(189,195)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,29)(22,220)(23,221)(24,222)(25,223)(26,224)(27,218)(28,219)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,96)(65,97)(66,98)(67,92)(68,93)(69,94)(70,95)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,120)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(127,152)(128,153)(129,154)(130,148)(131,149)(132,150)(133,151)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,177)(156,178)(157,179)(158,180)(159,181)(160,182)(161,176)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(183,208)(184,209)(185,210)(186,204)(187,205)(188,206)(189,207)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,39,77,8)(2,40,71,9)(3,41,72,10)(4,42,73,11)(5,36,74,12)(6,37,75,13)(7,38,76,14)(15,86,30,84)(16,87,31,78)(17,88,32,79)(18,89,33,80)(19,90,34,81)(20,91,35,82)(21,85,29,83)(22,61,220,52)(23,62,221,53)(24,63,222,54)(25,57,223,55)(26,58,224,56)(27,59,218,50)(28,60,219,51)(43,65,211,97)(44,66,212,98)(45,67,213,92)(46,68,214,93)(47,69,215,94)(48,70,216,95)(49,64,217,96)(99,177,121,155)(100,178,122,156)(101,179,123,157)(102,180,124,158)(103,181,125,159)(104,182,126,160)(105,176,120,161)(106,171,115,162)(107,172,116,163)(108,173,117,164)(109,174,118,165)(110,175,119,166)(111,169,113,167)(112,170,114,168)(127,208,152,183)(128,209,153,184)(129,210,154,185)(130,204,148,186)(131,205,149,187)(132,206,150,188)(133,207,151,189)(134,199,143,190)(135,200,144,191)(136,201,145,192)(137,202,146,193)(138,203,147,194)(139,197,141,195)(140,198,142,196), (1,141,85,151)(2,142,86,152)(3,143,87,153)(4,144,88,154)(5,145,89,148)(6,146,90,149)(7,147,91,150)(8,207,21,197)(9,208,15,198)(10,209,16,199)(11,210,17,200)(12,204,18,201)(13,205,19,202)(14,206,20,203)(22,173,211,155)(23,174,212,156)(24,175,213,157)(25,169,214,158)(26,170,215,159)(27,171,216,160)(28,172,217,161)(29,195,39,189)(30,196,40,183)(31,190,41,184)(32,191,42,185)(33,192,36,186)(34,193,37,187)(35,194,38,188)(43,177,220,164)(44,178,221,165)(45,179,222,166)(46,180,223,167)(47,181,224,168)(48,182,218,162)(49,176,219,163)(50,126,70,106)(51,120,64,107)(52,121,65,108)(53,122,66,109)(54,123,67,110)(55,124,68,111)(56,125,69,112)(57,102,93,113)(58,103,94,114)(59,104,95,115)(60,105,96,116)(61,99,97,117)(62,100,98,118)(63,101,92,119)(71,140,84,127)(72,134,78,128)(73,135,79,129)(74,136,80,130)(75,137,81,131)(76,138,82,132)(77,139,83,133)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,133)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,183)(29,160)(30,161)(31,155)(32,156)(33,157)(34,158)(35,159)(36,175)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,199)(44,200)(45,201)(46,202)(47,203)(48,197)(49,198)(50,115)(51,116)(52,117)(53,118)(54,119)(55,113)(56,114)(57,111)(58,112)(59,106)(60,107)(61,108)(62,109)(63,110)(64,105)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,152)(72,153)(73,154)(74,148)(75,149)(76,150)(77,151)(78,143)(79,144)(80,145)(81,146)(82,147)(83,141)(84,142)(85,139)(86,140)(87,134)(88,135)(89,136)(90,137)(91,138)(92,123)(93,124)(94,125)(95,126)(96,120)(97,121)(98,122)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(204,222)(205,223)(206,224)(207,218)(208,219)(209,220)(210,221), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(36,222)(37,223)(38,224)(39,218)(40,219)(41,220)(42,221)(50,77)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(64,84)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(99,143)(100,144)(101,145)(102,146)(103,147)(104,141)(105,142)(106,133)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,148)(120,140)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,189)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,204)(176,196)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(71,84)(72,78)(73,79)(74,80)(75,81)(76,82)(77,83)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(127,140)(128,134)(129,135)(130,136)(131,137)(132,138)(133,139)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(183,196)(184,190)(185,191)(186,192)(187,193)(188,194)(189,195)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,29)(22,220)(23,221)(24,222)(25,223)(26,224)(27,218)(28,219)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,96)(65,97)(66,98)(67,92)(68,93)(69,94)(70,95)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,120)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(127,152)(128,153)(129,154)(130,148)(131,149)(132,150)(133,151)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,177)(156,178)(157,179)(158,180)(159,181)(160,182)(161,176)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(183,208)(184,209)(185,210)(186,204)(187,205)(188,206)(189,207)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,39,77,8)(2,40,71,9)(3,41,72,10)(4,42,73,11)(5,36,74,12)(6,37,75,13)(7,38,76,14)(15,86,30,84)(16,87,31,78)(17,88,32,79)(18,89,33,80)(19,90,34,81)(20,91,35,82)(21,85,29,83)(22,61,220,52)(23,62,221,53)(24,63,222,54)(25,57,223,55)(26,58,224,56)(27,59,218,50)(28,60,219,51)(43,65,211,97)(44,66,212,98)(45,67,213,92)(46,68,214,93)(47,69,215,94)(48,70,216,95)(49,64,217,96)(99,177,121,155)(100,178,122,156)(101,179,123,157)(102,180,124,158)(103,181,125,159)(104,182,126,160)(105,176,120,161)(106,171,115,162)(107,172,116,163)(108,173,117,164)(109,174,118,165)(110,175,119,166)(111,169,113,167)(112,170,114,168)(127,208,152,183)(128,209,153,184)(129,210,154,185)(130,204,148,186)(131,205,149,187)(132,206,150,188)(133,207,151,189)(134,199,143,190)(135,200,144,191)(136,201,145,192)(137,202,146,193)(138,203,147,194)(139,197,141,195)(140,198,142,196), (1,141,85,151)(2,142,86,152)(3,143,87,153)(4,144,88,154)(5,145,89,148)(6,146,90,149)(7,147,91,150)(8,207,21,197)(9,208,15,198)(10,209,16,199)(11,210,17,200)(12,204,18,201)(13,205,19,202)(14,206,20,203)(22,173,211,155)(23,174,212,156)(24,175,213,157)(25,169,214,158)(26,170,215,159)(27,171,216,160)(28,172,217,161)(29,195,39,189)(30,196,40,183)(31,190,41,184)(32,191,42,185)(33,192,36,186)(34,193,37,187)(35,194,38,188)(43,177,220,164)(44,178,221,165)(45,179,222,166)(46,180,223,167)(47,181,224,168)(48,182,218,162)(49,176,219,163)(50,126,70,106)(51,120,64,107)(52,121,65,108)(53,122,66,109)(54,123,67,110)(55,124,68,111)(56,125,69,112)(57,102,93,113)(58,103,94,114)(59,104,95,115)(60,105,96,116)(61,99,97,117)(62,100,98,118)(63,101,92,119)(71,140,84,127)(72,134,78,128)(73,135,79,129)(74,136,80,130)(75,137,81,131)(76,138,82,132)(77,139,83,133) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,133),(2,127),(3,128),(4,129),(5,130),(6,131),(7,132),(8,162),(9,163),(10,164),(11,165),(12,166),(13,167),(14,168),(15,176),(16,177),(17,178),(18,179),(19,180),(20,181),(21,182),(22,184),(23,185),(24,186),(25,187),(26,188),(27,189),(28,183),(29,160),(30,161),(31,155),(32,156),(33,157),(34,158),(35,159),(36,175),(37,169),(38,170),(39,171),(40,172),(41,173),(42,174),(43,199),(44,200),(45,201),(46,202),(47,203),(48,197),(49,198),(50,115),(51,116),(52,117),(53,118),(54,119),(55,113),(56,114),(57,111),(58,112),(59,106),(60,107),(61,108),(62,109),(63,110),(64,105),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,152),(72,153),(73,154),(74,148),(75,149),(76,150),(77,151),(78,143),(79,144),(80,145),(81,146),(82,147),(83,141),(84,142),(85,139),(86,140),(87,134),(88,135),(89,136),(90,137),(91,138),(92,123),(93,124),(94,125),(95,126),(96,120),(97,121),(98,122),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(204,222),(205,223),(206,224),(207,218),(208,219),(209,220),(210,221)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,217),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(36,222),(37,223),(38,224),(39,218),(40,219),(41,220),(42,221),(50,77),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(64,84),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(99,143),(100,144),(101,145),(102,146),(103,147),(104,141),(105,142),(106,133),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,149),(114,150),(115,151),(116,152),(117,153),(118,154),(119,148),(120,140),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(155,199),(156,200),(157,201),(158,202),(159,203),(160,197),(161,198),(162,189),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188),(169,205),(170,206),(171,207),(172,208),(173,209),(174,210),(175,204),(176,196),(177,190),(178,191),(179,192),(180,193),(181,194),(182,195)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,217),(29,39),(30,40),(31,41),(32,42),(33,36),(34,37),(35,38),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,92),(71,84),(72,78),(73,79),(74,80),(75,81),(76,82),(77,83),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,126),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(127,140),(128,134),(129,135),(130,136),(131,137),(132,138),(133,139),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,182),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(183,196),(184,190),(185,191),(186,192),(187,193),(188,194),(189,195),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,77),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,39),(9,40),(10,41),(11,42),(12,36),(13,37),(14,38),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,29),(22,220),(23,221),(24,222),(25,223),(26,224),(27,218),(28,219),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,96),(65,97),(66,98),(67,92),(68,93),(69,94),(70,95),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,120),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(127,152),(128,153),(129,154),(130,148),(131,149),(132,150),(133,151),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(155,177),(156,178),(157,179),(158,180),(159,181),(160,182),(161,176),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(183,208),(184,209),(185,210),(186,204),(187,205),(188,206),(189,207),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198)], [(1,39,77,8),(2,40,71,9),(3,41,72,10),(4,42,73,11),(5,36,74,12),(6,37,75,13),(7,38,76,14),(15,86,30,84),(16,87,31,78),(17,88,32,79),(18,89,33,80),(19,90,34,81),(20,91,35,82),(21,85,29,83),(22,61,220,52),(23,62,221,53),(24,63,222,54),(25,57,223,55),(26,58,224,56),(27,59,218,50),(28,60,219,51),(43,65,211,97),(44,66,212,98),(45,67,213,92),(46,68,214,93),(47,69,215,94),(48,70,216,95),(49,64,217,96),(99,177,121,155),(100,178,122,156),(101,179,123,157),(102,180,124,158),(103,181,125,159),(104,182,126,160),(105,176,120,161),(106,171,115,162),(107,172,116,163),(108,173,117,164),(109,174,118,165),(110,175,119,166),(111,169,113,167),(112,170,114,168),(127,208,152,183),(128,209,153,184),(129,210,154,185),(130,204,148,186),(131,205,149,187),(132,206,150,188),(133,207,151,189),(134,199,143,190),(135,200,144,191),(136,201,145,192),(137,202,146,193),(138,203,147,194),(139,197,141,195),(140,198,142,196)], [(1,141,85,151),(2,142,86,152),(3,143,87,153),(4,144,88,154),(5,145,89,148),(6,146,90,149),(7,147,91,150),(8,207,21,197),(9,208,15,198),(10,209,16,199),(11,210,17,200),(12,204,18,201),(13,205,19,202),(14,206,20,203),(22,173,211,155),(23,174,212,156),(24,175,213,157),(25,169,214,158),(26,170,215,159),(27,171,216,160),(28,172,217,161),(29,195,39,189),(30,196,40,183),(31,190,41,184),(32,191,42,185),(33,192,36,186),(34,193,37,187),(35,194,38,188),(43,177,220,164),(44,178,221,165),(45,179,222,166),(46,180,223,167),(47,181,224,168),(48,182,218,162),(49,176,219,163),(50,126,70,106),(51,120,64,107),(52,121,65,108),(53,122,66,109),(54,123,67,110),(55,124,68,111),(56,125,69,112),(57,102,93,113),(58,103,94,114),(59,104,95,115),(60,105,96,116),(61,99,97,117),(62,100,98,118),(63,101,92,119),(71,140,84,127),(72,134,78,128),(73,135,79,129),(74,136,80,130),(75,137,81,131),(76,138,82,132),(77,139,83,133)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 28A | ··· | 28BT | 28BU | ··· | 28CR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C24.3C22 | C2×C4×C28 | C14×C22⋊C4 | C14×C4⋊C4 | D4×C2×C14 | D4×C14 | C24.3C22 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C2×D4 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 6 | 6 | 24 | 6 | 6 | 48 | 8 | 4 | 48 | 24 |
Matrix representation of C7×C24.3C22 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 20 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 17 | 28 | 0 | 0 |
| 0 | 27 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,20,0,0,0,0,0,20],[28,0,0,0,0,0,28,0,0,0,0,12,1,0,0,0,0,0,0,28,0,0,0,28,0],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[12,0,0,0,0,0,17,27,0,0,0,28,12,0,0,0,0,0,12,0,0,0,0,0,17],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,0] >;
C7×C24.3C22 in GAP, Magma, Sage, TeX
C_7\times C_2^4._3C_2^2
% in TeX
G:=Group("C7xC2^4.3C2^2"); // GroupNames label
G:=SmallGroup(448,798);
// by ID
G=gap.SmallGroup(448,798);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,604]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=1,f^2=e,g^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
// generators/relations