direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23.8Q8, C2.4(D4×C28), C22⋊C4⋊4C28, C14.105(C4×D4), (C2×C28).454D4, C23.8(C7×Q8), (C23×C28).7C2, (C23×C4).6C14, C23.35(C7×D4), C14.88C22≀C2, C23.15(C2×C28), C24.29(C2×C14), C22.34(D4×C14), (C22×C14).20Q8, C22.12(Q8×C14), C2.C42⋊7C14, (C22×C14).155D4, C14.83(C22⋊Q8), C23.58(C22×C14), C22.34(C22×C28), (C23×C14).86C22, (C22×C14).449C23, (C22×C28).493C22, C14.86(C22.D4), (C2×C4⋊C4)⋊2C14, (C2×C4)⋊2(C2×C28), C2.7(C14×C4⋊C4), C22⋊2(C7×C4⋊C4), (C14×C4⋊C4)⋊29C2, (C2×C14)⋊5(C4⋊C4), (C2×C28)⋊23(C2×C4), C14.62(C2×C4⋊C4), (C2×C4).99(C7×D4), (C7×C22⋊C4)⋊10C4, C2.2(C7×C22⋊Q8), C2.2(C7×C22≀C2), (C2×C14).601(C2×D4), (C2×C22⋊C4).5C14, (C2×C14).104(C2×Q8), C22.19(C7×C4○D4), (C14×C22⋊C4).11C2, (C22×C4).86(C2×C14), (C22×C14).82(C2×C4), (C2×C14).209(C4○D4), C2.2(C7×C22.D4), (C7×C2.C42)⋊23C2, (C2×C14).221(C22×C4), SmallGroup(448,793)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.8Q8
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=e2, 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=ce-1 >
Subgroups: 370 in 234 conjugacy classes, 106 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, 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, C23.8Q8, C7×C22⋊C4, 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×C23.8Q8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×C28, C7×D4, C7×Q8, C22×C14, C23.8Q8, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4⋊C4, D4×C28, C7×C22≀C2, C7×C22⋊Q8, C7×C22.D4, C7×C23.8Q8
►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 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(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)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 35)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 211)(16 212)(17 213)(18 214)(19 215)(20 216)(21 217)(22 48)(23 49)(24 43)(25 44)(26 45)(27 46)(28 47)(36 221)(37 222)(38 223)(39 224)(40 218)(41 219)(42 220)(57 83)(58 84)(59 78)(60 79)(61 80)(62 81)(63 82)(64 91)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 97)(72 98)(73 92)(74 93)(75 94)(76 95)(77 96)(99 125)(100 126)(101 120)(102 121)(103 122)(104 123)(105 124)(106 143)(107 144)(108 145)(109 146)(110 147)(111 141)(112 142)(113 139)(114 140)(115 134)(116 135)(117 136)(118 137)(119 138)(127 153)(128 154)(129 148)(130 149)(131 150)(132 151)(133 152)(155 181)(156 182)(157 176)(158 177)(159 178)(160 179)(161 180)(162 199)(163 200)(164 201)(165 202)(166 203)(167 197)(168 198)(169 195)(170 196)(171 190)(172 191)(173 192)(174 193)(175 194)(183 209)(184 210)(185 204)(186 205)(187 206)(188 207)(189 208)
(1 76)(2 77)(3 71)(4 72)(5 73)(6 74)(7 75)(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 95)(51 96)(52 97)(53 98)(54 92)(55 93)(56 94)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 82)(65 83)(66 84)(67 78)(68 79)(69 80)(70 81)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 151)(107 152)(108 153)(109 154)(110 148)(111 149)(112 150)(120 138)(121 139)(122 140)(123 134)(124 135)(125 136)(126 137)(127 145)(128 146)(129 147)(130 141)(131 142)(132 143)(133 144)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)
(1 134 78 143)(2 135 79 144)(3 136 80 145)(4 137 81 146)(5 138 82 147)(6 139 83 141)(7 140 84 142)(8 203 222 194)(9 197 223 195)(10 198 224 196)(11 199 218 190)(12 200 219 191)(13 201 220 192)(14 202 221 193)(15 209 24 155)(16 210 25 156)(17 204 26 157)(18 205 27 158)(19 206 28 159)(20 207 22 160)(21 208 23 161)(29 167 38 169)(30 168 39 170)(31 162 40 171)(32 163 41 172)(33 164 42 173)(34 165 36 174)(35 166 37 175)(43 181 211 183)(44 182 212 184)(45 176 213 185)(46 177 214 186)(47 178 215 187)(48 179 216 188)(49 180 217 189)(50 115 59 106)(51 116 60 107)(52 117 61 108)(53 118 62 109)(54 119 63 110)(55 113 57 111)(56 114 58 112)(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)(85 149 93 102)(86 150 94 103)(87 151 95 104)(88 152 96 105)(89 153 97 99)(90 154 98 100)(91 148 92 101)
(1 190 78 199)(2 191 79 200)(3 192 80 201)(4 193 81 202)(5 194 82 203)(6 195 83 197)(7 196 84 198)(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)(15 125 24 127)(16 126 25 128)(17 120 26 129)(18 121 27 130)(19 122 28 131)(20 123 22 132)(21 124 23 133)(29 139 38 141)(30 140 39 142)(31 134 40 143)(32 135 41 144)(33 136 42 145)(34 137 36 146)(35 138 37 147)(43 153 211 99)(44 154 212 100)(45 148 213 101)(46 149 214 102)(47 150 215 103)(48 151 216 104)(49 152 217 105)(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)(64 185 73 176)(65 186 74 177)(66 187 75 178)(67 188 76 179)(68 189 77 180)(69 183 71 181)(70 184 72 182)(85 205 93 158)(86 206 94 159)(87 207 95 160)(88 208 96 161)(89 209 97 155)(90 210 98 156)(91 204 92 157)
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)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(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)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,35)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,48)(23,49)(24,43)(25,44)(26,45)(27,46)(28,47)(36,221)(37,222)(38,223)(39,224)(40,218)(41,219)(42,220)(57,83)(58,84)(59,78)(60,79)(61,80)(62,81)(63,82)(64,91)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,97)(72,98)(73,92)(74,93)(75,94)(76,95)(77,96)(99,125)(100,126)(101,120)(102,121)(103,122)(104,123)(105,124)(106,143)(107,144)(108,145)(109,146)(110,147)(111,141)(112,142)(113,139)(114,140)(115,134)(116,135)(117,136)(118,137)(119,138)(127,153)(128,154)(129,148)(130,149)(131,150)(132,151)(133,152)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,199)(163,200)(164,201)(165,202)(166,203)(167,197)(168,198)(169,195)(170,196)(171,190)(172,191)(173,192)(174,193)(175,194)(183,209)(184,210)(185,204)(186,205)(187,206)(188,207)(189,208), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,82)(65,83)(66,84)(67,78)(68,79)(69,80)(70,81)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(120,138)(121,139)(122,140)(123,134)(124,135)(125,136)(126,137)(127,145)(128,146)(129,147)(130,141)(131,142)(132,143)(133,144)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,134,78,143)(2,135,79,144)(3,136,80,145)(4,137,81,146)(5,138,82,147)(6,139,83,141)(7,140,84,142)(8,203,222,194)(9,197,223,195)(10,198,224,196)(11,199,218,190)(12,200,219,191)(13,201,220,192)(14,202,221,193)(15,209,24,155)(16,210,25,156)(17,204,26,157)(18,205,27,158)(19,206,28,159)(20,207,22,160)(21,208,23,161)(29,167,38,169)(30,168,39,170)(31,162,40,171)(32,163,41,172)(33,164,42,173)(34,165,36,174)(35,166,37,175)(43,181,211,183)(44,182,212,184)(45,176,213,185)(46,177,214,186)(47,178,215,187)(48,179,216,188)(49,180,217,189)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(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)(85,149,93,102)(86,150,94,103)(87,151,95,104)(88,152,96,105)(89,153,97,99)(90,154,98,100)(91,148,92,101), (1,190,78,199)(2,191,79,200)(3,192,80,201)(4,193,81,202)(5,194,82,203)(6,195,83,197)(7,196,84,198)(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)(15,125,24,127)(16,126,25,128)(17,120,26,129)(18,121,27,130)(19,122,28,131)(20,123,22,132)(21,124,23,133)(29,139,38,141)(30,140,39,142)(31,134,40,143)(32,135,41,144)(33,136,42,145)(34,137,36,146)(35,138,37,147)(43,153,211,99)(44,154,212,100)(45,148,213,101)(46,149,214,102)(47,150,215,103)(48,151,216,104)(49,152,217,105)(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)(64,185,73,176)(65,186,74,177)(66,187,75,178)(67,188,76,179)(68,189,77,180)(69,183,71,181)(70,184,72,182)(85,205,93,158)(86,206,94,159)(87,207,95,160)(88,208,96,161)(89,209,97,155)(90,210,98,156)(91,204,92,157)>;
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)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(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)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,35)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,48)(23,49)(24,43)(25,44)(26,45)(27,46)(28,47)(36,221)(37,222)(38,223)(39,224)(40,218)(41,219)(42,220)(57,83)(58,84)(59,78)(60,79)(61,80)(62,81)(63,82)(64,91)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,97)(72,98)(73,92)(74,93)(75,94)(76,95)(77,96)(99,125)(100,126)(101,120)(102,121)(103,122)(104,123)(105,124)(106,143)(107,144)(108,145)(109,146)(110,147)(111,141)(112,142)(113,139)(114,140)(115,134)(116,135)(117,136)(118,137)(119,138)(127,153)(128,154)(129,148)(130,149)(131,150)(132,151)(133,152)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,199)(163,200)(164,201)(165,202)(166,203)(167,197)(168,198)(169,195)(170,196)(171,190)(172,191)(173,192)(174,193)(175,194)(183,209)(184,210)(185,204)(186,205)(187,206)(188,207)(189,208), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,82)(65,83)(66,84)(67,78)(68,79)(69,80)(70,81)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(120,138)(121,139)(122,140)(123,134)(124,135)(125,136)(126,137)(127,145)(128,146)(129,147)(130,141)(131,142)(132,143)(133,144)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,134,78,143)(2,135,79,144)(3,136,80,145)(4,137,81,146)(5,138,82,147)(6,139,83,141)(7,140,84,142)(8,203,222,194)(9,197,223,195)(10,198,224,196)(11,199,218,190)(12,200,219,191)(13,201,220,192)(14,202,221,193)(15,209,24,155)(16,210,25,156)(17,204,26,157)(18,205,27,158)(19,206,28,159)(20,207,22,160)(21,208,23,161)(29,167,38,169)(30,168,39,170)(31,162,40,171)(32,163,41,172)(33,164,42,173)(34,165,36,174)(35,166,37,175)(43,181,211,183)(44,182,212,184)(45,176,213,185)(46,177,214,186)(47,178,215,187)(48,179,216,188)(49,180,217,189)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(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)(85,149,93,102)(86,150,94,103)(87,151,95,104)(88,152,96,105)(89,153,97,99)(90,154,98,100)(91,148,92,101), (1,190,78,199)(2,191,79,200)(3,192,80,201)(4,193,81,202)(5,194,82,203)(6,195,83,197)(7,196,84,198)(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)(15,125,24,127)(16,126,25,128)(17,120,26,129)(18,121,27,130)(19,122,28,131)(20,123,22,132)(21,124,23,133)(29,139,38,141)(30,140,39,142)(31,134,40,143)(32,135,41,144)(33,136,42,145)(34,137,36,146)(35,138,37,147)(43,153,211,99)(44,154,212,100)(45,148,213,101)(46,149,214,102)(47,150,215,103)(48,151,216,104)(49,152,217,105)(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)(64,185,73,176)(65,186,74,177)(66,187,75,178)(67,188,76,179)(68,189,77,180)(69,183,71,181)(70,184,72,182)(85,205,93,158)(86,206,94,159)(87,207,95,160)(88,208,96,161)(89,209,97,155)(90,210,98,156)(91,204,92,157) );
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),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(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),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(176,194),(177,195),(178,196),(179,190),(180,191),(181,192),(182,193),(183,201),(184,202),(185,203),(186,197),(187,198),(188,199),(189,200)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,35),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,211),(16,212),(17,213),(18,214),(19,215),(20,216),(21,217),(22,48),(23,49),(24,43),(25,44),(26,45),(27,46),(28,47),(36,221),(37,222),(38,223),(39,224),(40,218),(41,219),(42,220),(57,83),(58,84),(59,78),(60,79),(61,80),(62,81),(63,82),(64,91),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,97),(72,98),(73,92),(74,93),(75,94),(76,95),(77,96),(99,125),(100,126),(101,120),(102,121),(103,122),(104,123),(105,124),(106,143),(107,144),(108,145),(109,146),(110,147),(111,141),(112,142),(113,139),(114,140),(115,134),(116,135),(117,136),(118,137),(119,138),(127,153),(128,154),(129,148),(130,149),(131,150),(132,151),(133,152),(155,181),(156,182),(157,176),(158,177),(159,178),(160,179),(161,180),(162,199),(163,200),(164,201),(165,202),(166,203),(167,197),(168,198),(169,195),(170,196),(171,190),(172,191),(173,192),(174,193),(175,194),(183,209),(184,210),(185,204),(186,205),(187,206),(188,207),(189,208)], [(1,76),(2,77),(3,71),(4,72),(5,73),(6,74),(7,75),(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(50,95),(51,96),(52,97),(53,98),(54,92),(55,93),(56,94),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,82),(65,83),(66,84),(67,78),(68,79),(69,80),(70,81),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,151),(107,152),(108,153),(109,154),(110,148),(111,149),(112,150),(120,138),(121,139),(122,140),(123,134),(124,135),(125,136),(126,137),(127,145),(128,146),(129,147),(130,141),(131,142),(132,143),(133,144),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(176,194),(177,195),(178,196),(179,190),(180,191),(181,192),(182,193),(183,201),(184,202),(185,203),(186,197),(187,198),(188,199),(189,200)], [(1,134,78,143),(2,135,79,144),(3,136,80,145),(4,137,81,146),(5,138,82,147),(6,139,83,141),(7,140,84,142),(8,203,222,194),(9,197,223,195),(10,198,224,196),(11,199,218,190),(12,200,219,191),(13,201,220,192),(14,202,221,193),(15,209,24,155),(16,210,25,156),(17,204,26,157),(18,205,27,158),(19,206,28,159),(20,207,22,160),(21,208,23,161),(29,167,38,169),(30,168,39,170),(31,162,40,171),(32,163,41,172),(33,164,42,173),(34,165,36,174),(35,166,37,175),(43,181,211,183),(44,182,212,184),(45,176,213,185),(46,177,214,186),(47,178,215,187),(48,179,216,188),(49,180,217,189),(50,115,59,106),(51,116,60,107),(52,117,61,108),(53,118,62,109),(54,119,63,110),(55,113,57,111),(56,114,58,112),(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),(85,149,93,102),(86,150,94,103),(87,151,95,104),(88,152,96,105),(89,153,97,99),(90,154,98,100),(91,148,92,101)], [(1,190,78,199),(2,191,79,200),(3,192,80,201),(4,193,81,202),(5,194,82,203),(6,195,83,197),(7,196,84,198),(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),(15,125,24,127),(16,126,25,128),(17,120,26,129),(18,121,27,130),(19,122,28,131),(20,123,22,132),(21,124,23,133),(29,139,38,141),(30,140,39,142),(31,134,40,143),(32,135,41,144),(33,136,42,145),(34,137,36,146),(35,138,37,147),(43,153,211,99),(44,154,212,100),(45,148,213,101),(46,149,214,102),(47,150,215,103),(48,151,216,104),(49,152,217,105),(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),(64,185,73,176),(65,186,74,177),(66,187,75,178),(67,188,76,179),(68,189,77,180),(69,183,71,181),(70,184,72,182),(85,205,93,158),(86,206,94,159),(87,207,95,160),(88,208,96,161),(89,209,97,155),(90,210,98,156),(91,204,92,157)]])
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×C23.8Q8 | C7×C2.C42 | C14×C22⋊C4 | C14×C4⋊C4 | C23×C28 | C7×C22⋊C4 | C23.8Q8 | 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×C23.8Q8 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 5 | 28 |
| 28 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 16 | 3 | 0 | 0 |
| 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 2 | 17 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,23,0,0,0,0,0,23],[28,0,0,0,0,0,1,0,0,0,0,13,28,0,0,0,0,0,1,5,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,1,0,0,0,0,13,28,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,16,2,0,0,0,3,13,0,0,0,0,0,12,2,0,0,0,1,17] >;
C7×C23.8Q8 in GAP, Magma, Sage, TeX
C_7\times C_2^3._8Q_8
% in TeX
G:=Group("C7xC2^3.8Q8"); // GroupNames label
G:=SmallGroup(448,793);
// by ID
G=gap.SmallGroup(448,793);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,2438]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=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=c*e^-1>;
// generators/relations