direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.81C23, C6.39(C4⋊Q8), (C2×C12).41Q8, (C2×C12).311D4, C22.74(C6×D4), C22.23(C6×Q8), C6.90(C22⋊Q8), C6.141(C4⋊D4), C23.85(C22×C6), C6.29(C42.C2), C2.C42.12C6, (C22×C6).462C23, (C22×C12).404C22, C6.92(C22.D4), C2.5(C3×C4⋊Q8), (C6×C4⋊C4).39C2, (C2×C4⋊C4).10C6, (C2×C4).4(C3×Q8), (C2×C4).18(C3×D4), C2.9(C3×C22⋊Q8), (C2×C6).614(C2×D4), C2.10(C3×C4⋊D4), (C2×C6).111(C2×Q8), C2.4(C3×C42.C2), (C22×C4).27(C2×C6), C22.41(C3×C4○D4), (C2×C6).222(C4○D4), C2.8(C3×C22.D4), (C3×C2.C42).28C2, SmallGroup(192,831)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.81C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >
Subgroups: 234 in 150 conjugacy classes, 78 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C23.81C23, C3×C2.C42, C3×C2.C42, C6×C4⋊C4, C6×C4⋊C4, C3×C23.81C23
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C6×D4, C6×Q8, C3×C4○D4, C23.81C23, C3×C4⋊D4, C3×C22⋊Q8, C3×C22.D4, C3×C42.C2, C3×C4⋊Q8, C3×C23.81C23
►Regular action on 192 points
Generators in S192
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 141 137)(6 142 138)(7 143 139)(8 144 140)(17 25 21)(18 26 22)(19 27 23)(20 28 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 49 45)(42 50 46)(43 51 47)(44 52 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 73 69)(66 74 70)(67 75 71)(68 76 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)(89 100 93)(90 97 94)(91 98 95)(92 99 96)(101 109 105)(102 110 106)(103 111 107)(104 112 108)(113 121 117)(114 122 118)(115 123 119)(116 124 120)(125 133 129)(126 134 130)(127 135 131)(128 136 132)(145 153 149)(146 154 150)(147 155 151)(148 156 152)(157 165 161)(158 166 162)(159 167 163)(160 168 164)(169 177 173)(170 178 174)(171 179 175)(172 180 176)(181 189 185)(182 190 186)(183 191 187)(184 192 188)
(1 103)(2 104)(3 101)(4 102)(5 49)(6 50)(7 51)(8 52)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 190)(98 191)(99 192)(100 189)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)(129 131)(130 132)(133 135)(134 136)(137 139)(138 140)(141 143)(142 144)(145 147)(146 148)(149 151)(150 152)(153 155)(154 156)(157 159)(158 160)(161 163)(162 164)(165 167)(166 168)(169 171)(170 172)(173 175)(174 176)(177 179)(178 180)(181 183)(182 184)(185 187)(186 188)(189 191)(190 192)
(1 147)(2 148)(3 145)(4 146)(5 100)(6 97)(7 98)(8 99)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)
(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)
(1 127 103 31)(2 32 104 128)(3 125 101 29)(4 30 102 126)(5 121 49 25)(6 26 50 122)(7 123 51 27)(8 28 52 124)(9 129 105 33)(10 34 106 130)(11 131 107 35)(12 36 108 132)(13 133 109 37)(14 38 110 134)(15 135 111 39)(16 40 112 136)(17 137 113 41)(18 42 114 138)(19 139 115 43)(20 44 116 140)(21 141 117 45)(22 46 118 142)(23 143 119 47)(24 48 120 144)(53 169 145 77)(54 78 146 170)(55 171 147 79)(56 80 148 172)(57 173 149 81)(58 82 150 174)(59 175 151 83)(60 84 152 176)(61 177 153 85)(62 86 154 178)(63 179 155 87)(64 88 156 180)(65 181 157 89)(66 90 158 182)(67 183 159 91)(68 92 160 184)(69 185 161 93)(70 94 162 186)(71 187 163 95)(72 96 164 188)(73 189 165 100)(74 97 166 190)(75 191 167 98)(76 99 168 192)
(1 115 103 19)(2 114 104 18)(3 113 101 17)(4 116 102 20)(5 85 49 177)(6 88 50 180)(7 87 51 179)(8 86 52 178)(9 117 105 21)(10 120 106 24)(11 119 107 23)(12 118 108 22)(13 121 109 25)(14 124 110 28)(15 123 111 27)(16 122 112 26)(29 89 125 181)(30 92 126 184)(31 91 127 183)(32 90 128 182)(33 93 129 185)(34 96 130 188)(35 95 131 187)(36 94 132 186)(37 100 133 189)(38 99 134 192)(39 98 135 191)(40 97 136 190)(41 169 137 77)(42 172 138 80)(43 171 139 79)(44 170 140 78)(45 173 141 81)(46 176 142 84)(47 175 143 83)(48 174 144 82)(53 157 145 65)(54 160 146 68)(55 159 147 67)(56 158 148 66)(57 161 149 69)(58 164 150 72)(59 163 151 71)(60 162 152 70)(61 165 153 73)(62 168 154 76)(63 167 155 75)(64 166 156 74)
G:=sub<Sym(192)| (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,141,137)(6,142,138)(7,143,139)(8,144,140)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,49,45)(42,50,46)(43,51,47)(44,52,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,73,69)(66,74,70)(67,75,71)(68,76,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84)(89,100,93)(90,97,94)(91,98,95)(92,99,96)(101,109,105)(102,110,106)(103,111,107)(104,112,108)(113,121,117)(114,122,118)(115,123,119)(116,124,120)(125,133,129)(126,134,130)(127,135,131)(128,136,132)(145,153,149)(146,154,150)(147,155,151)(148,156,152)(157,165,161)(158,166,162)(159,167,163)(160,168,164)(169,177,173)(170,178,174)(171,179,175)(172,180,176)(181,189,185)(182,190,186)(183,191,187)(184,192,188), (1,103)(2,104)(3,101)(4,102)(5,49)(6,50)(7,51)(8,52)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,190)(98,191)(99,192)(100,189), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,147)(2,148)(3,145)(4,146)(5,100)(6,97)(7,98)(8,99)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144), (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), (1,127,103,31)(2,32,104,128)(3,125,101,29)(4,30,102,126)(5,121,49,25)(6,26,50,122)(7,123,51,27)(8,28,52,124)(9,129,105,33)(10,34,106,130)(11,131,107,35)(12,36,108,132)(13,133,109,37)(14,38,110,134)(15,135,111,39)(16,40,112,136)(17,137,113,41)(18,42,114,138)(19,139,115,43)(20,44,116,140)(21,141,117,45)(22,46,118,142)(23,143,119,47)(24,48,120,144)(53,169,145,77)(54,78,146,170)(55,171,147,79)(56,80,148,172)(57,173,149,81)(58,82,150,174)(59,175,151,83)(60,84,152,176)(61,177,153,85)(62,86,154,178)(63,179,155,87)(64,88,156,180)(65,181,157,89)(66,90,158,182)(67,183,159,91)(68,92,160,184)(69,185,161,93)(70,94,162,186)(71,187,163,95)(72,96,164,188)(73,189,165,100)(74,97,166,190)(75,191,167,98)(76,99,168,192), (1,115,103,19)(2,114,104,18)(3,113,101,17)(4,116,102,20)(5,85,49,177)(6,88,50,180)(7,87,51,179)(8,86,52,178)(9,117,105,21)(10,120,106,24)(11,119,107,23)(12,118,108,22)(13,121,109,25)(14,124,110,28)(15,123,111,27)(16,122,112,26)(29,89,125,181)(30,92,126,184)(31,91,127,183)(32,90,128,182)(33,93,129,185)(34,96,130,188)(35,95,131,187)(36,94,132,186)(37,100,133,189)(38,99,134,192)(39,98,135,191)(40,97,136,190)(41,169,137,77)(42,172,138,80)(43,171,139,79)(44,170,140,78)(45,173,141,81)(46,176,142,84)(47,175,143,83)(48,174,144,82)(53,157,145,65)(54,160,146,68)(55,159,147,67)(56,158,148,66)(57,161,149,69)(58,164,150,72)(59,163,151,71)(60,162,152,70)(61,165,153,73)(62,168,154,76)(63,167,155,75)(64,166,156,74)>;
G:=Group( (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,141,137)(6,142,138)(7,143,139)(8,144,140)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,49,45)(42,50,46)(43,51,47)(44,52,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,73,69)(66,74,70)(67,75,71)(68,76,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84)(89,100,93)(90,97,94)(91,98,95)(92,99,96)(101,109,105)(102,110,106)(103,111,107)(104,112,108)(113,121,117)(114,122,118)(115,123,119)(116,124,120)(125,133,129)(126,134,130)(127,135,131)(128,136,132)(145,153,149)(146,154,150)(147,155,151)(148,156,152)(157,165,161)(158,166,162)(159,167,163)(160,168,164)(169,177,173)(170,178,174)(171,179,175)(172,180,176)(181,189,185)(182,190,186)(183,191,187)(184,192,188), (1,103)(2,104)(3,101)(4,102)(5,49)(6,50)(7,51)(8,52)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,190)(98,191)(99,192)(100,189), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,147)(2,148)(3,145)(4,146)(5,100)(6,97)(7,98)(8,99)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144), (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), (1,127,103,31)(2,32,104,128)(3,125,101,29)(4,30,102,126)(5,121,49,25)(6,26,50,122)(7,123,51,27)(8,28,52,124)(9,129,105,33)(10,34,106,130)(11,131,107,35)(12,36,108,132)(13,133,109,37)(14,38,110,134)(15,135,111,39)(16,40,112,136)(17,137,113,41)(18,42,114,138)(19,139,115,43)(20,44,116,140)(21,141,117,45)(22,46,118,142)(23,143,119,47)(24,48,120,144)(53,169,145,77)(54,78,146,170)(55,171,147,79)(56,80,148,172)(57,173,149,81)(58,82,150,174)(59,175,151,83)(60,84,152,176)(61,177,153,85)(62,86,154,178)(63,179,155,87)(64,88,156,180)(65,181,157,89)(66,90,158,182)(67,183,159,91)(68,92,160,184)(69,185,161,93)(70,94,162,186)(71,187,163,95)(72,96,164,188)(73,189,165,100)(74,97,166,190)(75,191,167,98)(76,99,168,192), (1,115,103,19)(2,114,104,18)(3,113,101,17)(4,116,102,20)(5,85,49,177)(6,88,50,180)(7,87,51,179)(8,86,52,178)(9,117,105,21)(10,120,106,24)(11,119,107,23)(12,118,108,22)(13,121,109,25)(14,124,110,28)(15,123,111,27)(16,122,112,26)(29,89,125,181)(30,92,126,184)(31,91,127,183)(32,90,128,182)(33,93,129,185)(34,96,130,188)(35,95,131,187)(36,94,132,186)(37,100,133,189)(38,99,134,192)(39,98,135,191)(40,97,136,190)(41,169,137,77)(42,172,138,80)(43,171,139,79)(44,170,140,78)(45,173,141,81)(46,176,142,84)(47,175,143,83)(48,174,144,82)(53,157,145,65)(54,160,146,68)(55,159,147,67)(56,158,148,66)(57,161,149,69)(58,164,150,72)(59,163,151,71)(60,162,152,70)(61,165,153,73)(62,168,154,76)(63,167,155,75)(64,166,156,74) );
G=PermutationGroup([[(1,15,11),(2,16,12),(3,13,9),(4,14,10),(5,141,137),(6,142,138),(7,143,139),(8,144,140),(17,25,21),(18,26,22),(19,27,23),(20,28,24),(29,37,33),(30,38,34),(31,39,35),(32,40,36),(41,49,45),(42,50,46),(43,51,47),(44,52,48),(53,61,57),(54,62,58),(55,63,59),(56,64,60),(65,73,69),(66,74,70),(67,75,71),(68,76,72),(77,85,81),(78,86,82),(79,87,83),(80,88,84),(89,100,93),(90,97,94),(91,98,95),(92,99,96),(101,109,105),(102,110,106),(103,111,107),(104,112,108),(113,121,117),(114,122,118),(115,123,119),(116,124,120),(125,133,129),(126,134,130),(127,135,131),(128,136,132),(145,153,149),(146,154,150),(147,155,151),(148,156,152),(157,165,161),(158,166,162),(159,167,163),(160,168,164),(169,177,173),(170,178,174),(171,179,175),(172,180,176),(181,189,185),(182,190,186),(183,191,187),(184,192,188)], [(1,103),(2,104),(3,101),(4,102),(5,49),(6,50),(7,51),(8,52),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,169),(78,170),(79,171),(80,172),(81,173),(82,174),(83,175),(84,176),(85,177),(86,178),(87,179),(88,180),(89,181),(90,182),(91,183),(92,184),(93,185),(94,186),(95,187),(96,188),(97,190),(98,191),(99,192),(100,189)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96),(97,99),(98,100),(101,103),(102,104),(105,107),(106,108),(109,111),(110,112),(113,115),(114,116),(117,119),(118,120),(121,123),(122,124),(125,127),(126,128),(129,131),(130,132),(133,135),(134,136),(137,139),(138,140),(141,143),(142,144),(145,147),(146,148),(149,151),(150,152),(153,155),(154,156),(157,159),(158,160),(161,163),(162,164),(165,167),(166,168),(169,171),(170,172),(173,175),(174,176),(177,179),(178,180),(181,183),(182,184),(185,187),(186,188),(189,191),(190,192)], [(1,147),(2,148),(3,145),(4,146),(5,100),(6,97),(7,98),(8,99),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,165),(26,166),(27,167),(28,168),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144)], [(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)], [(1,127,103,31),(2,32,104,128),(3,125,101,29),(4,30,102,126),(5,121,49,25),(6,26,50,122),(7,123,51,27),(8,28,52,124),(9,129,105,33),(10,34,106,130),(11,131,107,35),(12,36,108,132),(13,133,109,37),(14,38,110,134),(15,135,111,39),(16,40,112,136),(17,137,113,41),(18,42,114,138),(19,139,115,43),(20,44,116,140),(21,141,117,45),(22,46,118,142),(23,143,119,47),(24,48,120,144),(53,169,145,77),(54,78,146,170),(55,171,147,79),(56,80,148,172),(57,173,149,81),(58,82,150,174),(59,175,151,83),(60,84,152,176),(61,177,153,85),(62,86,154,178),(63,179,155,87),(64,88,156,180),(65,181,157,89),(66,90,158,182),(67,183,159,91),(68,92,160,184),(69,185,161,93),(70,94,162,186),(71,187,163,95),(72,96,164,188),(73,189,165,100),(74,97,166,190),(75,191,167,98),(76,99,168,192)], [(1,115,103,19),(2,114,104,18),(3,113,101,17),(4,116,102,20),(5,85,49,177),(6,88,50,180),(7,87,51,179),(8,86,52,178),(9,117,105,21),(10,120,106,24),(11,119,107,23),(12,118,108,22),(13,121,109,25),(14,124,110,28),(15,123,111,27),(16,122,112,26),(29,89,125,181),(30,92,126,184),(31,91,127,183),(32,90,128,182),(33,93,129,185),(34,96,130,188),(35,95,131,187),(36,94,132,186),(37,100,133,189),(38,99,134,192),(39,98,135,191),(40,97,136,190),(41,169,137,77),(42,172,138,80),(43,171,139,79),(44,170,140,78),(45,173,141,81),(46,176,142,84),(47,175,143,83),(48,174,144,82),(53,157,145,65),(54,160,146,68),(55,159,147,67),(56,158,148,66),(57,161,149,69),(58,164,150,72),(59,163,151,71),(60,162,152,70),(61,165,153,73),(62,168,154,76),(63,167,155,75),(64,166,156,74)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4N | 6A | ··· | 6N | 12A | ··· | 12AB |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.81C23 | C3×C2.C42 | C6×C4⋊C4 | C23.81C23 | C2.C42 | C2×C4⋊C4 | C2×C12 | C2×C12 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 3 | 4 | 2 | 6 | 8 | 4 | 4 | 6 | 8 | 8 | 12 |
Matrix representation of C3×C23.81C23 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 0 | 0 | 0 | 0 | 12 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 9 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,10,1,0,0,0,0,0,0,7,12,0,0,0,0,11,6],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,6,9,0,0,0,0,6,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,11,5,0,0,0,0,0,0,12,6,0,0,0,0,0,1] >;
C3×C23.81C23 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{81}C_2^3 % in TeX
G:=Group("C3xC2^3.81C2^3"); // GroupNames label
G:=SmallGroup(192,831);
// by ID
G=gap.SmallGroup(192,831);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1094,1059,142]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=c,f^2=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,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
// generators/relations