direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6.C42, C24.85D6, C23.64D12, C23.19Dic6, (C22×C12)⋊9C4, (C23×C4).8S3, C6⋊(C2.C42), (C2×C6).27C42, (C23×C12).1C2, C6.27(C2×C42), C23.69(C4×S3), (C22×C4)⋊7Dic3, (C22×C6).24Q8, (C22×Dic3)⋊9C4, (C22×C4).420D6, (C22×C6).190D4, C22.59(C2×D12), C22.53(D6⋊C4), (C23×C6).95C22, (C23×Dic3).6C2, C23.47(C2×Dic3), C22.22(C4×Dic3), C22.30(C2×Dic6), C23.106(C3⋊D4), (C22×C6).359C23, C23.309(C22×S3), C22.23(C4⋊Dic3), C22.27(Dic3⋊C4), (C22×C12).480C22, C22.25(C22×Dic3), C22.32(C6.D4), (C22×Dic3).194C22, C6.54(C2×C4⋊C4), C2.3(C2×D6⋊C4), (C2×C12)⋊34(C2×C4), (C2×C4)⋊9(C2×Dic3), C22.64(S3×C2×C4), C2.3(C2×C4⋊Dic3), (C2×C6).42(C2×Q8), C2.15(C2×C4×Dic3), (C2×C6).55(C4⋊C4), (C2×C6).545(C2×D4), C6.64(C2×C22⋊C4), C2.3(C2×Dic3⋊C4), C3⋊2(C2×C2.C42), (C2×Dic3)⋊20(C2×C4), (C22×C6).97(C2×C4), C2.2(C2×C6.D4), C22.83(C2×C3⋊D4), (C2×C6).69(C22⋊C4), (C2×C6).140(C22×C4), SmallGroup(192,767)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6.C42
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >
Subgroups: 664 in 330 conjugacy classes, 183 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, C2×C6, C2×C6, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C2.C42, C23×C4, C23×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23×C6, C2×C2.C42, C6.C42, C23×Dic3, C23×C12, C2×C6.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C22×S3, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C2.C42, C6.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, C2×C6.D4, C2×C6.C42
►Regular action on 192 points
Generators in S192
(1 93)(2 94)(3 95)(4 96)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 73)(21 74)(22 75)(23 76)(24 77)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 144)(34 133)(35 134)(36 135)(37 117)(38 118)(39 119)(40 120)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 97)(61 160)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 157)(71 158)(72 159)(121 153)(122 154)(123 155)(124 156)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(169 188)(170 189)(171 190)(172 191)(173 192)(174 181)(175 182)(176 183)(177 184)(178 185)(179 186)(180 187)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 177)(14 178)(15 179)(16 180)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 145)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 85)(59 86)(60 87)(73 191)(74 192)(75 181)(76 182)(77 183)(78 184)(79 185)(80 186)(81 187)(82 188)(83 189)(84 190)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 143)(122 144)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)
(1 71)(2 72)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 104)(38 105)(39 106)(40 107)(41 108)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 109)(60 110)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 133)(83 134)(84 135)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 168)(92 157)(93 158)(94 159)(95 160)(96 161)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 181)(129 182)(130 183)(131 184)(132 185)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 177)(152 178)(153 179)(154 180)(155 169)(156 170)
(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)
(1 145 42 13)(2 176 43 35)(3 155 44 23)(4 174 45 33)(5 153 46 21)(6 172 47 31)(7 151 48 19)(8 170 37 29)(9 149 38 17)(10 180 39 27)(11 147 40 15)(12 178 41 25)(14 64 146 102)(16 62 148 100)(18 72 150 98)(20 70 152 108)(22 68 154 106)(24 66 156 104)(26 63 179 101)(28 61 169 99)(30 71 171 97)(32 69 173 107)(34 67 175 105)(36 65 177 103)(49 83 159 130)(50 139 160 188)(51 81 161 128)(52 137 162 186)(53 79 163 126)(54 135 164 184)(55 77 165 124)(56 133 166 182)(57 75 167 122)(58 143 168 192)(59 73 157 132)(60 141 158 190)(74 85 121 114)(76 95 123 112)(78 93 125 110)(80 91 127 120)(82 89 129 118)(84 87 131 116)(86 191 115 142)(88 189 117 140)(90 187 119 138)(92 185 109 136)(94 183 111 134)(96 181 113 144)
G:=sub<Sym(192)| (1,93)(2,94)(3,95)(4,96)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,133)(35,134)(36,135)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,97)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,157)(71,158)(72,159)(121,153)(122,154)(123,155)(124,156)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(169,188)(170,189)(171,190)(172,191)(173,192)(174,181)(175,182)(176,183)(177,184)(178,185)(179,186)(180,187), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,177)(14,178)(15,179)(16,180)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,145)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87)(73,191)(74,192)(75,181)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,104)(38,105)(39,106)(40,107)(41,108)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,109)(60,110)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,133)(83,134)(84,135)(85,162)(86,163)(87,164)(88,165)(89,166)(90,167)(91,168)(92,157)(93,158)(94,159)(95,160)(96,161)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,181)(129,182)(130,183)(131,184)(132,185)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,169)(156,170), (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), (1,145,42,13)(2,176,43,35)(3,155,44,23)(4,174,45,33)(5,153,46,21)(6,172,47,31)(7,151,48,19)(8,170,37,29)(9,149,38,17)(10,180,39,27)(11,147,40,15)(12,178,41,25)(14,64,146,102)(16,62,148,100)(18,72,150,98)(20,70,152,108)(22,68,154,106)(24,66,156,104)(26,63,179,101)(28,61,169,99)(30,71,171,97)(32,69,173,107)(34,67,175,105)(36,65,177,103)(49,83,159,130)(50,139,160,188)(51,81,161,128)(52,137,162,186)(53,79,163,126)(54,135,164,184)(55,77,165,124)(56,133,166,182)(57,75,167,122)(58,143,168,192)(59,73,157,132)(60,141,158,190)(74,85,121,114)(76,95,123,112)(78,93,125,110)(80,91,127,120)(82,89,129,118)(84,87,131,116)(86,191,115,142)(88,189,117,140)(90,187,119,138)(92,185,109,136)(94,183,111,134)(96,181,113,144)>;
G:=Group( (1,93)(2,94)(3,95)(4,96)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,133)(35,134)(36,135)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,97)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,157)(71,158)(72,159)(121,153)(122,154)(123,155)(124,156)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(169,188)(170,189)(171,190)(172,191)(173,192)(174,181)(175,182)(176,183)(177,184)(178,185)(179,186)(180,187), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,177)(14,178)(15,179)(16,180)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,145)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87)(73,191)(74,192)(75,181)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192), (1,71)(2,72)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,104)(38,105)(39,106)(40,107)(41,108)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,109)(60,110)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,133)(83,134)(84,135)(85,162)(86,163)(87,164)(88,165)(89,166)(90,167)(91,168)(92,157)(93,158)(94,159)(95,160)(96,161)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,181)(129,182)(130,183)(131,184)(132,185)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,169)(156,170), (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), (1,145,42,13)(2,176,43,35)(3,155,44,23)(4,174,45,33)(5,153,46,21)(6,172,47,31)(7,151,48,19)(8,170,37,29)(9,149,38,17)(10,180,39,27)(11,147,40,15)(12,178,41,25)(14,64,146,102)(16,62,148,100)(18,72,150,98)(20,70,152,108)(22,68,154,106)(24,66,156,104)(26,63,179,101)(28,61,169,99)(30,71,171,97)(32,69,173,107)(34,67,175,105)(36,65,177,103)(49,83,159,130)(50,139,160,188)(51,81,161,128)(52,137,162,186)(53,79,163,126)(54,135,164,184)(55,77,165,124)(56,133,166,182)(57,75,167,122)(58,143,168,192)(59,73,157,132)(60,141,158,190)(74,85,121,114)(76,95,123,112)(78,93,125,110)(80,91,127,120)(82,89,129,118)(84,87,131,116)(86,191,115,142)(88,189,117,140)(90,187,119,138)(92,185,109,136)(94,183,111,134)(96,181,113,144) );
G=PermutationGroup([[(1,93),(2,94),(3,95),(4,96),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,73),(21,74),(22,75),(23,76),(24,77),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,144),(34,133),(35,134),(36,135),(37,117),(38,118),(39,119),(40,120),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,97),(61,160),(62,161),(63,162),(64,163),(65,164),(66,165),(67,166),(68,167),(69,168),(70,157),(71,158),(72,159),(121,153),(122,154),(123,155),(124,156),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(169,188),(170,189),(171,190),(172,191),(173,192),(174,181),(175,182),(176,183),(177,184),(178,185),(179,186),(180,187)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,177),(14,178),(15,179),(16,180),(17,169),(18,170),(19,171),(20,172),(21,173),(22,174),(23,175),(24,176),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,145),(37,72),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,85),(59,86),(60,87),(73,191),(74,192),(75,181),(76,182),(77,183),(78,184),(79,185),(80,186),(81,187),(82,188),(83,189),(84,190),(109,163),(110,164),(111,165),(112,166),(113,167),(114,168),(115,157),(116,158),(117,159),(118,160),(119,161),(120,162),(121,143),(122,144),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(131,141),(132,142)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192)], [(1,71),(2,72),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29),(37,104),(38,105),(39,106),(40,107),(41,108),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,109),(60,110),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,133),(83,134),(84,135),(85,162),(86,163),(87,164),(88,165),(89,166),(90,167),(91,168),(92,157),(93,158),(94,159),(95,160),(96,161),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,181),(129,182),(130,183),(131,184),(132,185),(145,171),(146,172),(147,173),(148,174),(149,175),(150,176),(151,177),(152,178),(153,179),(154,180),(155,169),(156,170)], [(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)], [(1,145,42,13),(2,176,43,35),(3,155,44,23),(4,174,45,33),(5,153,46,21),(6,172,47,31),(7,151,48,19),(8,170,37,29),(9,149,38,17),(10,180,39,27),(11,147,40,15),(12,178,41,25),(14,64,146,102),(16,62,148,100),(18,72,150,98),(20,70,152,108),(22,68,154,106),(24,66,156,104),(26,63,179,101),(28,61,169,99),(30,71,171,97),(32,69,173,107),(34,67,175,105),(36,65,177,103),(49,83,159,130),(50,139,160,188),(51,81,161,128),(52,137,162,186),(53,79,163,126),(54,135,164,184),(55,77,165,124),(56,133,166,182),(57,75,167,122),(58,143,168,192),(59,73,157,132),(60,141,158,190),(74,85,121,114),(76,95,123,112),(78,93,125,110),(80,91,127,120),(82,89,129,118),(84,87,131,116),(86,191,115,142),(88,189,117,140),(90,187,119,138),(92,185,109,136),(94,183,111,134),(96,181,113,144)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3 | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C2×C6.C42 | C6.C42 | C23×Dic3 | C23×C12 | C22×Dic3 | C22×C12 | C23×C4 | C22×C6 | C22×C6 | C22×C4 | C22×C4 | C24 | C23 | C23 | C23 | C23 |
| # reps | 1 | 4 | 2 | 1 | 16 | 8 | 1 | 6 | 2 | 4 | 2 | 1 | 4 | 8 | 4 | 8 |
Matrix representation of C2×C6.C42 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,5,5,0,0,0,0,8,0],[8,0,0,0,0,0,0,1,0,0,0,0,0,0,10,10,0,0,0,0,7,3,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;
C2×C6.C42 in GAP, Magma, Sage, TeX
C_2\times C_6.C_4^2
% in TeX
G:=Group("C2xC6.C4^2"); // GroupNames label
G:=SmallGroup(192,767);
// by ID
G=gap.SmallGroup(192,767);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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=d*e^5>;
// generators/relations