direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23.7Q8, (C22×C4)⋊8C28, C28⋊9(C22⋊C4), (C22×C28)⋊19C4, (C2×C28).511D4, C23.7(C7×Q8), (C23×C4).9C14, C23.21(C7×D4), C22.9(Q8×C14), (C23×C28).22C2, C23.25(C2×C28), C24.27(C2×C14), C22.30(D4×C14), (C22×C14).19Q8, C2.C42⋊1C14, (C22×C14).126D4, C14.82(C22⋊Q8), C14.132(C4⋊D4), (C23×C14).84C22, C23.53(C22×C14), C22.29(C22×C28), C14.53(C42⋊C2), (C22×C28).573C22, (C22×C14).444C23, (C2×C4⋊C4)⋊1C14, C2.4(C14×C4⋊C4), C22⋊1(C7×C4⋊C4), C4⋊2(C7×C22⋊C4), (C14×C4⋊C4)⋊28C2, (C2×C14)⋊4(C4⋊C4), C14.59(C2×C4⋊C4), C2.1(C7×C4⋊D4), (C2×C4).55(C2×C28), C2.1(C7×C22⋊Q8), (C2×C4).116(C7×D4), C2.5(C14×C22⋊C4), (C2×C28).330(C2×C4), (C2×C14).597(C2×D4), (C2×C22⋊C4).3C14, (C14×C22⋊C4).9C2, C14.92(C2×C22⋊C4), C2.5(C7×C42⋊C2), (C2×C14).101(C2×Q8), C22.15(C7×C4○D4), (C7×C2.C42)⋊3C2, (C22×C4).84(C2×C14), (C2×C14).205(C4○D4), (C22×C14).111(C2×C4), (C2×C14).216(C22×C4), SmallGroup(448,788)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.7Q8
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 370 in 234 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.7Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C22×C28, C23×C14, C7×C2.C42, C14×C22⋊C4, C14×C4⋊C4, C23×C28, C7×C23.7Q8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C23.7Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C22⋊C4, C14×C4⋊C4, C7×C42⋊C2, C7×C4⋊D4, C7×C22⋊Q8, C7×C23.7Q8
►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)
(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 25)(16 26)(17 27)(18 28)(19 22)(20 23)(21 24)(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)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 190)(176 186)(177 187)(178 188)(179 189)(180 183)(181 184)(182 185)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 49)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 216)(30 217)(31 211)(32 212)(33 213)(34 214)(35 215)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(99 125)(100 126)(101 120)(102 121)(103 122)(104 123)(105 124)(106 151)(107 152)(108 153)(109 154)(110 148)(111 149)(112 150)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(155 181)(156 182)(157 176)(158 177)(159 178)(160 179)(161 180)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 25)(16 26)(17 27)(18 28)(19 22)(20 23)(21 24)(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 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 78)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(64 74)(65 75)(66 76)(67 77)(68 71)(69 72)(70 73)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 134)(120 130)(121 131)(122 132)(123 133)(124 127)(125 128)(126 129)(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)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 190)(176 186)(177 187)(178 188)(179 189)(180 183)(181 184)(182 185)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 115 59 151)(2 116 60 152)(3 117 61 153)(4 118 62 154)(5 119 63 148)(6 113 57 149)(7 114 58 150)(8 204 28 175)(9 205 22 169)(10 206 23 170)(11 207 24 171)(12 208 25 172)(13 209 26 173)(14 210 27 174)(15 161 217 198)(16 155 211 199)(17 156 212 200)(18 157 213 201)(19 158 214 202)(20 159 215 203)(21 160 216 197)(29 191 48 179)(30 192 49 180)(31 193 43 181)(32 194 44 182)(33 195 45 176)(34 196 46 177)(35 190 47 178)(36 166 222 186)(37 167 223 187)(38 168 224 188)(39 162 218 189)(40 163 219 183)(41 164 220 184)(42 165 221 185)(50 133 77 106)(51 127 71 107)(52 128 72 108)(53 129 73 109)(54 130 74 110)(55 131 75 111)(56 132 76 112)(64 139 83 120)(65 140 84 121)(66 134 78 122)(67 135 79 123)(68 136 80 124)(69 137 81 125)(70 138 82 126)(85 104 95 141)(86 105 96 142)(87 99 97 143)(88 100 98 144)(89 101 92 145)(90 102 93 146)(91 103 94 147)
(1 189 77 207)(2 183 71 208)(3 184 72 209)(4 185 73 210)(5 186 74 204)(6 187 75 205)(7 188 76 206)(8 119 222 110)(9 113 223 111)(10 114 224 112)(11 115 218 106)(12 116 219 107)(13 117 220 108)(14 118 221 109)(15 142 30 124)(16 143 31 125)(17 144 32 126)(18 145 33 120)(19 146 34 121)(20 147 35 122)(21 141 29 123)(22 149 37 131)(23 150 38 132)(24 151 39 133)(25 152 40 127)(26 153 41 128)(27 154 42 129)(28 148 36 130)(43 137 211 99)(44 138 212 100)(45 139 213 101)(46 140 214 102)(47 134 215 103)(48 135 216 104)(49 136 217 105)(50 171 59 162)(51 172 60 163)(52 173 61 164)(53 174 62 165)(54 175 63 166)(55 169 57 167)(56 170 58 168)(64 201 89 176)(65 202 90 177)(66 203 91 178)(67 197 85 179)(68 198 86 180)(69 199 87 181)(70 200 88 182)(78 159 94 190)(79 160 95 191)(80 161 96 192)(81 155 97 193)(82 156 98 194)(83 157 92 195)(84 158 93 196)
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), (8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(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)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,49)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,216)(30,217)(31,211)(32,212)(33,213)(34,214)(35,215)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(99,125)(100,126)(101,120)(102,121)(103,122)(104,123)(105,124)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(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)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,78)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,134)(120,130)(121,131)(122,132)(123,133)(124,127)(125,128)(126,129)(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)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,115,59,151)(2,116,60,152)(3,117,61,153)(4,118,62,154)(5,119,63,148)(6,113,57,149)(7,114,58,150)(8,204,28,175)(9,205,22,169)(10,206,23,170)(11,207,24,171)(12,208,25,172)(13,209,26,173)(14,210,27,174)(15,161,217,198)(16,155,211,199)(17,156,212,200)(18,157,213,201)(19,158,214,202)(20,159,215,203)(21,160,216,197)(29,191,48,179)(30,192,49,180)(31,193,43,181)(32,194,44,182)(33,195,45,176)(34,196,46,177)(35,190,47,178)(36,166,222,186)(37,167,223,187)(38,168,224,188)(39,162,218,189)(40,163,219,183)(41,164,220,184)(42,165,221,185)(50,133,77,106)(51,127,71,107)(52,128,72,108)(53,129,73,109)(54,130,74,110)(55,131,75,111)(56,132,76,112)(64,139,83,120)(65,140,84,121)(66,134,78,122)(67,135,79,123)(68,136,80,124)(69,137,81,125)(70,138,82,126)(85,104,95,141)(86,105,96,142)(87,99,97,143)(88,100,98,144)(89,101,92,145)(90,102,93,146)(91,103,94,147), (1,189,77,207)(2,183,71,208)(3,184,72,209)(4,185,73,210)(5,186,74,204)(6,187,75,205)(7,188,76,206)(8,119,222,110)(9,113,223,111)(10,114,224,112)(11,115,218,106)(12,116,219,107)(13,117,220,108)(14,118,221,109)(15,142,30,124)(16,143,31,125)(17,144,32,126)(18,145,33,120)(19,146,34,121)(20,147,35,122)(21,141,29,123)(22,149,37,131)(23,150,38,132)(24,151,39,133)(25,152,40,127)(26,153,41,128)(27,154,42,129)(28,148,36,130)(43,137,211,99)(44,138,212,100)(45,139,213,101)(46,140,214,102)(47,134,215,103)(48,135,216,104)(49,136,217,105)(50,171,59,162)(51,172,60,163)(52,173,61,164)(53,174,62,165)(54,175,63,166)(55,169,57,167)(56,170,58,168)(64,201,89,176)(65,202,90,177)(66,203,91,178)(67,197,85,179)(68,198,86,180)(69,199,87,181)(70,200,88,182)(78,159,94,190)(79,160,95,191)(80,161,96,192)(81,155,97,193)(82,156,98,194)(83,157,92,195)(84,158,93,196)>;
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), (8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(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)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,49)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,216)(30,217)(31,211)(32,212)(33,213)(34,214)(35,215)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(99,125)(100,126)(101,120)(102,121)(103,122)(104,123)(105,124)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(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)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,78)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,134)(120,130)(121,131)(122,132)(123,133)(124,127)(125,128)(126,129)(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)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,115,59,151)(2,116,60,152)(3,117,61,153)(4,118,62,154)(5,119,63,148)(6,113,57,149)(7,114,58,150)(8,204,28,175)(9,205,22,169)(10,206,23,170)(11,207,24,171)(12,208,25,172)(13,209,26,173)(14,210,27,174)(15,161,217,198)(16,155,211,199)(17,156,212,200)(18,157,213,201)(19,158,214,202)(20,159,215,203)(21,160,216,197)(29,191,48,179)(30,192,49,180)(31,193,43,181)(32,194,44,182)(33,195,45,176)(34,196,46,177)(35,190,47,178)(36,166,222,186)(37,167,223,187)(38,168,224,188)(39,162,218,189)(40,163,219,183)(41,164,220,184)(42,165,221,185)(50,133,77,106)(51,127,71,107)(52,128,72,108)(53,129,73,109)(54,130,74,110)(55,131,75,111)(56,132,76,112)(64,139,83,120)(65,140,84,121)(66,134,78,122)(67,135,79,123)(68,136,80,124)(69,137,81,125)(70,138,82,126)(85,104,95,141)(86,105,96,142)(87,99,97,143)(88,100,98,144)(89,101,92,145)(90,102,93,146)(91,103,94,147), (1,189,77,207)(2,183,71,208)(3,184,72,209)(4,185,73,210)(5,186,74,204)(6,187,75,205)(7,188,76,206)(8,119,222,110)(9,113,223,111)(10,114,224,112)(11,115,218,106)(12,116,219,107)(13,117,220,108)(14,118,221,109)(15,142,30,124)(16,143,31,125)(17,144,32,126)(18,145,33,120)(19,146,34,121)(20,147,35,122)(21,141,29,123)(22,149,37,131)(23,150,38,132)(24,151,39,133)(25,152,40,127)(26,153,41,128)(27,154,42,129)(28,148,36,130)(43,137,211,99)(44,138,212,100)(45,139,213,101)(46,140,214,102)(47,134,215,103)(48,135,216,104)(49,136,217,105)(50,171,59,162)(51,172,60,163)(52,173,61,164)(53,174,62,165)(54,175,63,166)(55,169,57,167)(56,170,58,168)(64,201,89,176)(65,202,90,177)(66,203,91,178)(67,197,85,179)(68,198,86,180)(69,199,87,181)(70,200,88,182)(78,159,94,190)(79,160,95,191)(80,161,96,192)(81,155,97,193)(82,156,98,194)(83,157,92,195)(84,158,93,196) );
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)], [(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,25),(16,26),(17,27),(18,28),(19,22),(20,23),(21,24),(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),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,190),(176,186),(177,187),(178,188),(179,189),(180,183),(181,184),(182,185),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,49),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,216),(30,217),(31,211),(32,212),(33,213),(34,214),(35,215),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(99,125),(100,126),(101,120),(102,121),(103,122),(104,123),(105,124),(106,151),(107,152),(108,153),(109,154),(110,148),(111,149),(112,150),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,147),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(155,181),(156,182),(157,176),(158,177),(159,178),(160,179),(161,180),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,25),(16,26),(17,27),(18,28),(19,22),(20,23),(21,24),(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),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,78),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,92),(64,74),(65,75),(66,76),(67,77),(68,71),(69,72),(70,73),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,134),(120,130),(121,131),(122,132),(123,133),(124,127),(125,128),(126,129),(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),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,190),(176,186),(177,187),(178,188),(179,189),(180,183),(181,184),(182,185),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,115,59,151),(2,116,60,152),(3,117,61,153),(4,118,62,154),(5,119,63,148),(6,113,57,149),(7,114,58,150),(8,204,28,175),(9,205,22,169),(10,206,23,170),(11,207,24,171),(12,208,25,172),(13,209,26,173),(14,210,27,174),(15,161,217,198),(16,155,211,199),(17,156,212,200),(18,157,213,201),(19,158,214,202),(20,159,215,203),(21,160,216,197),(29,191,48,179),(30,192,49,180),(31,193,43,181),(32,194,44,182),(33,195,45,176),(34,196,46,177),(35,190,47,178),(36,166,222,186),(37,167,223,187),(38,168,224,188),(39,162,218,189),(40,163,219,183),(41,164,220,184),(42,165,221,185),(50,133,77,106),(51,127,71,107),(52,128,72,108),(53,129,73,109),(54,130,74,110),(55,131,75,111),(56,132,76,112),(64,139,83,120),(65,140,84,121),(66,134,78,122),(67,135,79,123),(68,136,80,124),(69,137,81,125),(70,138,82,126),(85,104,95,141),(86,105,96,142),(87,99,97,143),(88,100,98,144),(89,101,92,145),(90,102,93,146),(91,103,94,147)], [(1,189,77,207),(2,183,71,208),(3,184,72,209),(4,185,73,210),(5,186,74,204),(6,187,75,205),(7,188,76,206),(8,119,222,110),(9,113,223,111),(10,114,224,112),(11,115,218,106),(12,116,219,107),(13,117,220,108),(14,118,221,109),(15,142,30,124),(16,143,31,125),(17,144,32,126),(18,145,33,120),(19,146,34,121),(20,147,35,122),(21,141,29,123),(22,149,37,131),(23,150,38,132),(24,151,39,133),(25,152,40,127),(26,153,41,128),(27,154,42,129),(28,148,36,130),(43,137,211,99),(44,138,212,100),(45,139,213,101),(46,140,214,102),(47,134,215,103),(48,135,216,104),(49,136,217,105),(50,171,59,162),(51,172,60,163),(52,173,61,164),(53,174,62,165),(54,175,63,166),(55,169,57,167),(56,170,58,168),(64,201,89,176),(65,202,90,177),(66,203,91,178),(67,197,85,179),(68,198,86,180),(69,199,87,181),(70,200,88,182),(78,159,94,190),(79,160,95,191),(80,161,96,192),(81,155,97,193),(82,156,98,194),(83,157,92,195),(84,158,93,196)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 28A | ··· | 28AV | 28AW | ··· | 28CR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | Q8 | C4○D4 | C7×D4 | C7×D4 | C7×Q8 | C7×C4○D4 |
| kernel | C7×C23.7Q8 | C7×C2.C42 | C14×C22⋊C4 | C14×C4⋊C4 | C23×C28 | C22×C28 | C23.7Q8 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C2×C28 | C22×C14 | C22×C14 | C2×C14 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 6 | 12 | 12 | 12 | 6 | 48 | 4 | 2 | 2 | 4 | 24 | 12 | 12 | 24 |
Matrix representation of C7×C23.7Q8 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 17 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 1 | 1 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 28 | 28 | 0 | 0 |
| 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 17 |
| 0 | 0 | 0 | 20 | 26 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,16,0,0,0,0,0,16],[28,0,0,0,0,0,1,0,0,0,0,1,28,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,17,0,0,0,0,17,12,0,0,0,0,0,28,1,0,0,0,27,1],[17,0,0,0,0,0,28,2,0,0,0,28,1,0,0,0,0,0,3,20,0,0,0,17,26] >;
C7×C23.7Q8 in GAP, Magma, Sage, TeX
C_7\times C_2^3._7Q_8
% in TeX
G:=Group("C7xC2^3.7Q8"); // GroupNames label
G:=SmallGroup(448,788);
// by ID
G=gap.SmallGroup(448,788);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,400,2438]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations