direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.47C24, C14.1642+ 1+4, (C4×D4)⋊20C14, (D4×C28)⋊49C2, C4⋊D4⋊15C14, C42.C2⋊9C14, C42⋊2C2⋊6C14, C42.47(C2×C14), C42⋊C2⋊16C14, C28.324(C4○D4), (C4×C28).288C22, (C2×C14).373C24, (C2×C28).963C23, (D4×C14).221C22, C22.D4⋊11C14, C23.19(C22×C14), C22.47(C23×C14), C2.16(C7×2+ 1+4), (C22×C28).458C22, (C22×C14).102C23, (C2×C4⋊C4)⋊22C14, (C14×C4⋊C4)⋊49C2, C4.36(C7×C4○D4), (C7×C4⋊D4)⋊42C2, C4⋊C4.73(C2×C14), C2.26(C14×C4○D4), (C2×D4).34(C2×C14), C14.245(C2×C4○D4), (C7×C42.C2)⋊26C2, C22.11(C7×C4○D4), (C7×C42⋊C2)⋊37C2, (C7×C42⋊2C2)⋊17C2, C22⋊C4.23(C2×C14), (C7×C4⋊C4).399C22, (C22×C4).12(C2×C14), (C2×C4).62(C22×C14), (C2×C14).179(C4○D4), (C7×C22.D4)⋊30C2, (C7×C22⋊C4).155C22, SmallGroup(448,1336)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22.47C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 362 in 238 conjugacy classes, 150 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.47C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, C14×C4⋊C4, C7×C42⋊C2, D4×C28, D4×C28, C7×C4⋊D4, C7×C4⋊D4, C7×C22.D4, C7×C42.C2, C7×C42⋊2C2, C7×C22.47C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, C22×C14, C22.47C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×C22.47C24
►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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 132)(114 133)(115 127)(116 128)(117 129)(118 130)(119 131)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 188)(170 189)(171 183)(172 184)(173 185)(174 186)(175 187)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 216)(198 217)(199 211)(200 212)(201 213)(202 214)(203 215)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 131)(107 132)(108 133)(109 127)(110 128)(111 129)(112 130)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 159)(135 160)(136 161)(137 155)(138 156)(139 157)(140 158)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 187)(163 188)(164 189)(165 183)(166 184)(167 185)(168 186)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 215)(191 216)(192 217)(193 211)(194 212)(195 213)(196 214)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 148)(7 149)(8 118)(9 119)(10 113)(11 114)(12 115)(13 116)(14 117)(15 125)(16 126)(17 120)(18 121)(19 122)(20 123)(21 124)(22 129)(23 130)(24 131)(25 132)(26 133)(27 127)(28 128)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 138)(37 139)(38 140)(39 134)(40 135)(41 136)(42 137)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 103 47 94)(2 104 48 95)(3 105 49 96)(4 99 43 97)(5 100 44 98)(6 101 45 92)(7 102 46 93)(8 186 224 177)(9 187 218 178)(10 188 219 179)(11 189 220 180)(12 183 221 181)(13 184 222 182)(14 185 223 176)(15 165 27 171)(16 166 28 172)(17 167 22 173)(18 168 23 174)(19 162 24 175)(20 163 25 169)(21 164 26 170)(29 79 40 85)(30 80 41 86)(31 81 42 87)(32 82 36 88)(33 83 37 89)(34 84 38 90)(35 78 39 91)(50 119 63 106)(51 113 57 107)(52 114 58 108)(53 115 59 109)(54 116 60 110)(55 117 61 111)(56 118 62 112)(64 129 73 120)(65 130 74 121)(66 131 75 122)(67 132 76 123)(68 133 77 124)(69 127 71 125)(70 128 72 126)(134 215 147 206)(135 216 141 207)(136 217 142 208)(137 211 143 209)(138 212 144 210)(139 213 145 204)(140 214 146 205)(148 201 157 195)(149 202 158 196)(150 203 159 190)(151 197 160 191)(152 198 161 192)(153 199 155 193)(154 200 156 194)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 132)(114 133)(115 127)(116 128)(117 129)(118 130)(119 131)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(148 157)(149 158)(150 159)(151 160)(152 161)(153 155)(154 156)(162 175)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(176 185)(177 186)(178 187)(179 188)(180 189)(181 183)(182 184)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 213)(205 214)(206 215)(207 216)(208 217)(209 211)(210 212)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 214 23 202)(9 215 24 203)(10 216 25 197)(11 217 26 198)(12 211 27 199)(13 212 28 200)(14 213 22 201)(15 193 221 209)(16 194 222 210)(17 195 223 204)(18 196 224 205)(19 190 218 206)(20 191 219 207)(21 192 220 208)(29 76 48 57)(30 77 49 58)(31 71 43 59)(32 72 44 60)(33 73 45 61)(34 74 46 62)(35 75 47 63)(78 106 94 122)(79 107 95 123)(80 108 96 124)(81 109 97 125)(82 110 98 126)(83 111 92 120)(84 112 93 121)(85 113 104 132)(86 114 105 133)(87 115 99 127)(88 116 100 128)(89 117 101 129)(90 118 102 130)(91 119 103 131)(134 178 150 162)(135 179 151 163)(136 180 152 164)(137 181 153 165)(138 182 154 166)(139 176 148 167)(140 177 149 168)(141 188 160 169)(142 189 161 170)(143 183 155 171)(144 184 156 172)(145 185 157 173)(146 186 158 174)(147 187 159 175)
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,159)(135,160)(136,161)(137,155)(138,156)(139,157)(140,158)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,187)(163,188)(164,189)(165,183)(166,184)(167,185)(168,186)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,215)(191,216)(192,217)(193,211)(194,212)(195,213)(196,214)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,129)(23,130)(24,131)(25,132)(26,133)(27,127)(28,128)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,103,47,94)(2,104,48,95)(3,105,49,96)(4,99,43,97)(5,100,44,98)(6,101,45,92)(7,102,46,93)(8,186,224,177)(9,187,218,178)(10,188,219,179)(11,189,220,180)(12,183,221,181)(13,184,222,182)(14,185,223,176)(15,165,27,171)(16,166,28,172)(17,167,22,173)(18,168,23,174)(19,162,24,175)(20,163,25,169)(21,164,26,170)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91)(50,119,63,106)(51,113,57,107)(52,114,58,108)(53,115,59,109)(54,116,60,110)(55,117,61,111)(56,118,62,112)(64,129,73,120)(65,130,74,121)(66,131,75,122)(67,132,76,123)(68,133,77,124)(69,127,71,125)(70,128,72,126)(134,215,147,206)(135,216,141,207)(136,217,142,208)(137,211,143,209)(138,212,144,210)(139,213,145,204)(140,214,146,205)(148,201,157,195)(149,202,158,196)(150,203,159,190)(151,197,160,191)(152,198,161,192)(153,199,155,193)(154,200,156,194), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,157)(149,158)(150,159)(151,160)(152,161)(153,155)(154,156)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,185)(177,186)(178,187)(179,188)(180,189)(181,183)(182,184)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,213)(205,214)(206,215)(207,216)(208,217)(209,211)(210,212), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,214,23,202)(9,215,24,203)(10,216,25,197)(11,217,26,198)(12,211,27,199)(13,212,28,200)(14,213,22,201)(15,193,221,209)(16,194,222,210)(17,195,223,204)(18,196,224,205)(19,190,218,206)(20,191,219,207)(21,192,220,208)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,106,94,122)(79,107,95,123)(80,108,96,124)(81,109,97,125)(82,110,98,126)(83,111,92,120)(84,112,93,121)(85,113,104,132)(86,114,105,133)(87,115,99,127)(88,116,100,128)(89,117,101,129)(90,118,102,130)(91,119,103,131)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,188,160,169)(142,189,161,170)(143,183,155,171)(144,184,156,172)(145,185,157,173)(146,186,158,174)(147,187,159,175)>;
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,159)(135,160)(136,161)(137,155)(138,156)(139,157)(140,158)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,187)(163,188)(164,189)(165,183)(166,184)(167,185)(168,186)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,215)(191,216)(192,217)(193,211)(194,212)(195,213)(196,214)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,129)(23,130)(24,131)(25,132)(26,133)(27,127)(28,128)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,103,47,94)(2,104,48,95)(3,105,49,96)(4,99,43,97)(5,100,44,98)(6,101,45,92)(7,102,46,93)(8,186,224,177)(9,187,218,178)(10,188,219,179)(11,189,220,180)(12,183,221,181)(13,184,222,182)(14,185,223,176)(15,165,27,171)(16,166,28,172)(17,167,22,173)(18,168,23,174)(19,162,24,175)(20,163,25,169)(21,164,26,170)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91)(50,119,63,106)(51,113,57,107)(52,114,58,108)(53,115,59,109)(54,116,60,110)(55,117,61,111)(56,118,62,112)(64,129,73,120)(65,130,74,121)(66,131,75,122)(67,132,76,123)(68,133,77,124)(69,127,71,125)(70,128,72,126)(134,215,147,206)(135,216,141,207)(136,217,142,208)(137,211,143,209)(138,212,144,210)(139,213,145,204)(140,214,146,205)(148,201,157,195)(149,202,158,196)(150,203,159,190)(151,197,160,191)(152,198,161,192)(153,199,155,193)(154,200,156,194), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,157)(149,158)(150,159)(151,160)(152,161)(153,155)(154,156)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,185)(177,186)(178,187)(179,188)(180,189)(181,183)(182,184)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,213)(205,214)(206,215)(207,216)(208,217)(209,211)(210,212), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,214,23,202)(9,215,24,203)(10,216,25,197)(11,217,26,198)(12,211,27,199)(13,212,28,200)(14,213,22,201)(15,193,221,209)(16,194,222,210)(17,195,223,204)(18,196,224,205)(19,190,218,206)(20,191,219,207)(21,192,220,208)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,106,94,122)(79,107,95,123)(80,108,96,124)(81,109,97,125)(82,110,98,126)(83,111,92,120)(84,112,93,121)(85,113,104,132)(86,114,105,133)(87,115,99,127)(88,116,100,128)(89,117,101,129)(90,118,102,130)(91,119,103,131)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,188,160,169)(142,189,161,170)(143,183,155,171)(144,184,156,172)(145,185,157,173)(146,186,158,174)(147,187,159,175) );
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)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,132),(114,133),(115,127),(116,128),(117,129),(118,130),(119,131),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,160),(142,161),(143,155),(144,156),(145,157),(146,158),(147,159),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,188),(170,189),(171,183),(172,184),(173,185),(174,186),(175,187),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,216),(198,217),(199,211),(200,212),(201,213),(202,214),(203,215)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,43),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,103),(79,104),(80,105),(81,99),(82,100),(83,101),(84,102),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,131),(107,132),(108,133),(109,127),(110,128),(111,129),(112,130),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,159),(135,160),(136,161),(137,155),(138,156),(139,157),(140,158),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(162,187),(163,188),(164,189),(165,183),(166,184),(167,185),(168,186),(169,179),(170,180),(171,181),(172,182),(173,176),(174,177),(175,178),(190,215),(191,216),(192,217),(193,211),(194,212),(195,213),(196,214),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,148),(7,149),(8,118),(9,119),(10,113),(11,114),(12,115),(13,116),(14,117),(15,125),(16,126),(17,120),(18,121),(19,122),(20,123),(21,124),(22,129),(23,130),(24,131),(25,132),(26,133),(27,127),(28,128),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,138),(37,139),(38,140),(39,134),(40,135),(41,136),(42,137),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,103,47,94),(2,104,48,95),(3,105,49,96),(4,99,43,97),(5,100,44,98),(6,101,45,92),(7,102,46,93),(8,186,224,177),(9,187,218,178),(10,188,219,179),(11,189,220,180),(12,183,221,181),(13,184,222,182),(14,185,223,176),(15,165,27,171),(16,166,28,172),(17,167,22,173),(18,168,23,174),(19,162,24,175),(20,163,25,169),(21,164,26,170),(29,79,40,85),(30,80,41,86),(31,81,42,87),(32,82,36,88),(33,83,37,89),(34,84,38,90),(35,78,39,91),(50,119,63,106),(51,113,57,107),(52,114,58,108),(53,115,59,109),(54,116,60,110),(55,117,61,111),(56,118,62,112),(64,129,73,120),(65,130,74,121),(66,131,75,122),(67,132,76,123),(68,133,77,124),(69,127,71,125),(70,128,72,126),(134,215,147,206),(135,216,141,207),(136,217,142,208),(137,211,143,209),(138,212,144,210),(139,213,145,204),(140,214,146,205),(148,201,157,195),(149,202,158,196),(150,203,159,190),(151,197,160,191),(152,198,161,192),(153,199,155,193),(154,200,156,194)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,224),(9,218),(10,219),(11,220),(12,221),(13,222),(14,223),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,132),(114,133),(115,127),(116,128),(117,129),(118,130),(119,131),(134,147),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(148,157),(149,158),(150,159),(151,160),(152,161),(153,155),(154,156),(162,175),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174),(176,185),(177,186),(178,187),(179,188),(180,189),(181,183),(182,184),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,213),(205,214),(206,215),(207,216),(208,217),(209,211),(210,212)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,214,23,202),(9,215,24,203),(10,216,25,197),(11,217,26,198),(12,211,27,199),(13,212,28,200),(14,213,22,201),(15,193,221,209),(16,194,222,210),(17,195,223,204),(18,196,224,205),(19,190,218,206),(20,191,219,207),(21,192,220,208),(29,76,48,57),(30,77,49,58),(31,71,43,59),(32,72,44,60),(33,73,45,61),(34,74,46,62),(35,75,47,63),(78,106,94,122),(79,107,95,123),(80,108,96,124),(81,109,97,125),(82,110,98,126),(83,111,92,120),(84,112,93,121),(85,113,104,132),(86,114,105,133),(87,115,99,127),(88,116,100,128),(89,117,101,129),(90,118,102,130),(91,119,103,131),(134,178,150,162),(135,179,151,163),(136,180,152,164),(137,181,153,165),(138,182,154,166),(139,176,148,167),(140,177,149,168),(141,188,160,169),(142,189,161,170),(143,183,155,171),(144,184,156,172),(145,185,157,173),(146,186,158,174),(147,187,159,175)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AV | 28A | ··· | 28BH | 28BI | ··· | 28CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C4○D4 | C4○D4 | C7×C4○D4 | C7×C4○D4 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×C22.47C24 | C14×C4⋊C4 | C7×C42⋊C2 | D4×C28 | C7×C4⋊D4 | C7×C22.D4 | C7×C42.C2 | C7×C42⋊2C2 | C22.47C24 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4⋊D4 | C22.D4 | C42.C2 | C42⋊2C2 | C28 | C2×C14 | C4 | C22 | C14 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 6 | 6 | 6 | 24 | 24 | 12 | 6 | 12 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×C22.47C24 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 17 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,12,0,0,17,0,0,0,0,0,0,1,0,0,1,0],[17,0,0,0,0,12,0,0,0,0,17,0,0,0,0,17],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[0,28,0,0,1,0,0,0,0,0,28,0,0,0,0,28] >;
C7×C22.47C24 in GAP, Magma, Sage, TeX
C_7\times C_2^2._{47}C_2^4 % in TeX
G:=Group("C7xC2^2.47C2^4"); // GroupNames label
G:=SmallGroup(448,1336);
// by ID
G=gap.SmallGroup(448,1336);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,2360,4790,1690,416]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations