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G = C3×C23.78C23  order 192 = 26·3

Direct product of C3 and C23.78C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C23.78C23, (C2×C12)⋊8Q8, C6.38(C4⋊Q8), C6.93C22≀C2, (C2×C12).308D4, C22.71(C6×D4), (C22×Q8).9C6, C22.21(C6×Q8), C6.88(C22⋊Q8), C23.82(C22×C6), C2.C42.11C6, (C22×C6).459C23, (C22×C12).403C22, (C2×C4)⋊1(C3×Q8), C2.4(C3×C4⋊Q8), (C2×C4⋊C4).9C6, (C6×C4⋊C4).38C2, (Q8×C2×C6).12C2, (C2×C4).15(C3×D4), C2.7(C3×C22⋊Q8), C2.7(C3×C22≀C2), (C2×C6).611(C2×D4), (C2×C6).109(C2×Q8), (C22×C4).26(C2×C6), C22.38(C3×C4○D4), (C2×C6).219(C4○D4), (C3×C2.C42).27C2, SmallGroup(192,828)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C3×C23.78C23
Chief series
C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — Q8×C2×C6 — C3×C23.78C23
Lower central
C1 — C23 — C3×C23.78C23
Upper central
C1 — C22×C6 — C3×C23.78C23

Generators and relations for C3×C23.78C23
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=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: 282 in 182 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C12, C2×C12, C3×Q8, C22×C6, C2.C42, C2×C4⋊C4, C22×Q8, C3×C4⋊C4, C22×C12, C22×C12, C6×Q8, C23.78C23, C3×C2.C42, C6×C4⋊C4, Q8×C2×C6, C3×C23.78C23
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22≀C2, C22⋊Q8, C4⋊Q8, C6×D4, C6×Q8, C3×C4○D4, C23.78C23, C3×C22≀C2, C3×C22⋊Q8, C3×C4⋊Q8, C3×C23.78C23

Smallest permutation representation of C3×C23.78C23
►Regular action on 192 points
Generators in S192
(1 109 105)(2 110 106)(3 111 107)(4 112 108)(5 192 188)(6 189 185)(7 190 186)(8 191 187)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 29 25)(22 30 26)(23 31 27)(24 32 28)(33 41 37)(34 42 38)(35 43 39)(36 44 40)(45 53 49)(46 54 50)(47 55 51)(48 56 52)(57 65 61)(58 66 62)(59 67 63)(60 68 64)(69 77 73)(70 78 74)(71 79 75)(72 80 76)(81 89 85)(82 90 86)(83 91 87)(84 92 88)(93 102 98)(94 103 99)(95 104 100)(96 101 97)(113 121 117)(114 122 118)(115 123 119)(116 124 120)(125 133 129)(126 134 130)(127 135 131)(128 136 132)(137 145 141)(138 146 142)(139 147 143)(140 148 144)(149 157 153)(150 158 154)(151 159 155)(152 160 156)(161 169 165)(162 170 166)(163 171 167)(164 172 168)(173 181 177)(174 182 178)(175 183 179)(176 184 180)

(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 9)(2 10)(3 11)(4 12)(5 103)(6 104)(7 101)(8 102)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 151)(58 152)(59 149)(60 150)(61 155)(62 156)(63 153)(64 154)(65 159)(66 160)(67 157)(68 158)(69 163)(70 164)(71 161)(72 162)(73 167)(74 168)(75 165)(76 166)(77 171)(78 172)(79 169)(80 170)(81 175)(82 176)(83 173)(84 174)(85 179)(86 180)(87 177)(88 178)(89 183)(90 184)(91 181)(92 182)(93 187)(94 188)(95 185)(96 186)(97 190)(98 191)(99 192)(100 189)

(1 149)(2 150)(3 151)(4 152)(5 148)(6 145)(7 146)(8 147)(9 59)(10 60)(11 57)(12 58)(13 63)(14 64)(15 61)(16 62)(17 67)(18 68)(19 65)(20 66)(21 71)(22 72)(23 69)(24 70)(25 75)(26 76)(27 73)(28 74)(29 79)(30 80)(31 77)(32 78)(33 83)(34 84)(35 81)(36 82)(37 87)(38 88)(39 85)(40 86)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)(49 100)(50 97)(51 98)(52 99)(53 104)(54 101)(55 102)(56 103)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 181)(134 182)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 192)

(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 3 125)(2 126 4 128)(5 170 7 172)(6 169 8 171)(9 35 11 33)(10 34 12 36)(13 39 15 37)(14 38 16 40)(17 43 19 41)(18 42 20 44)(21 47 23 45)(22 46 24 48)(25 51 27 49)(26 50 28 52)(29 55 31 53)(30 54 32 56)(57 83 59 81)(58 82 60 84)(61 87 63 85)(62 86 64 88)(65 91 67 89)(66 90 68 92)(69 95 71 93)(70 94 72 96)(73 100 75 98)(74 99 76 97)(77 104 79 102)(78 103 80 101)(105 131 107 129)(106 130 108 132)(109 135 111 133)(110 134 112 136)(113 139 115 137)(114 138 116 140)(117 143 119 141)(118 142 120 144)(121 147 123 145)(122 146 124 148)(149 175 151 173)(150 174 152 176)(153 179 155 177)(154 178 156 180)(157 183 159 181)(158 182 160 184)(161 187 163 185)(162 186 164 188)(165 191 167 189)(166 190 168 192)

(1 115 3 113)(2 24 4 22)(5 44 7 42)(6 133 8 135)(9 23 11 21)(10 116 12 114)(13 27 15 25)(14 120 16 118)(17 31 19 29)(18 124 20 122)(26 106 28 108)(30 110 32 112)(33 93 35 95)(34 188 36 186)(37 98 39 100)(38 192 40 190)(41 102 43 104)(45 83 47 81)(46 174 48 176)(49 87 51 85)(50 178 52 180)(53 91 55 89)(54 182 56 184)(57 71 59 69)(58 162 60 164)(61 75 63 73)(62 166 64 168)(65 79 67 77)(66 170 68 172)(70 152 72 150)(74 156 76 154)(78 160 80 158)(82 138 84 140)(86 142 88 144)(90 146 92 148)(94 128 96 126)(97 130 99 132)(101 134 103 136)(105 119 107 117)(109 123 111 121)(125 187 127 185)(129 191 131 189)(137 173 139 175)(141 177 143 179)(145 181 147 183)(149 163 151 161)(153 167 155 165)(157 171 159 169)
 
G:=sub<Sym(192)| (1,109,105)(2,110,106)(3,111,107)(4,112,108)(5,192,188)(6,189,185)(7,190,186)(8,191,187)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(33,41,37)(34,42,38)(35,43,39)(36,44,40)(45,53,49)(46,54,50)(47,55,51)(48,56,52)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(69,77,73)(70,78,74)(71,79,75)(72,80,76)(81,89,85)(82,90,86)(83,91,87)(84,92,88)(93,102,98)(94,103,99)(95,104,100)(96,101,97)(113,121,117)(114,122,118)(115,123,119)(116,124,120)(125,133,129)(126,134,130)(127,135,131)(128,136,132)(137,145,141)(138,146,142)(139,147,143)(140,148,144)(149,157,153)(150,158,154)(151,159,155)(152,160,156)(161,169,165)(162,170,166)(163,171,167)(164,172,168)(173,181,177)(174,182,178)(175,183,179)(176,184,180), (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), (1,9)(2,10)(3,11)(4,12)(5,103)(6,104)(7,101)(8,102)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,151)(58,152)(59,149)(60,150)(61,155)(62,156)(63,153)(64,154)(65,159)(66,160)(67,157)(68,158)(69,163)(70,164)(71,161)(72,162)(73,167)(74,168)(75,165)(76,166)(77,171)(78,172)(79,169)(80,170)(81,175)(82,176)(83,173)(84,174)(85,179)(86,180)(87,177)(88,178)(89,183)(90,184)(91,181)(92,182)(93,187)(94,188)(95,185)(96,186)(97,190)(98,191)(99,192)(100,189), (1,149)(2,150)(3,151)(4,152)(5,148)(6,145)(7,146)(8,147)(9,59)(10,60)(11,57)(12,58)(13,63)(14,64)(15,61)(16,62)(17,67)(18,68)(19,65)(20,66)(21,71)(22,72)(23,69)(24,70)(25,75)(26,76)(27,73)(28,74)(29,79)(30,80)(31,77)(32,78)(33,83)(34,84)(35,81)(36,82)(37,87)(38,88)(39,85)(40,86)(41,91)(42,92)(43,89)(44,90)(45,95)(46,96)(47,93)(48,94)(49,100)(50,97)(51,98)(52,99)(53,104)(54,101)(55,102)(56,103)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192), (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), (1,127,3,125)(2,126,4,128)(5,170,7,172)(6,169,8,171)(9,35,11,33)(10,34,12,36)(13,39,15,37)(14,38,16,40)(17,43,19,41)(18,42,20,44)(21,47,23,45)(22,46,24,48)(25,51,27,49)(26,50,28,52)(29,55,31,53)(30,54,32,56)(57,83,59,81)(58,82,60,84)(61,87,63,85)(62,86,64,88)(65,91,67,89)(66,90,68,92)(69,95,71,93)(70,94,72,96)(73,100,75,98)(74,99,76,97)(77,104,79,102)(78,103,80,101)(105,131,107,129)(106,130,108,132)(109,135,111,133)(110,134,112,136)(113,139,115,137)(114,138,116,140)(117,143,119,141)(118,142,120,144)(121,147,123,145)(122,146,124,148)(149,175,151,173)(150,174,152,176)(153,179,155,177)(154,178,156,180)(157,183,159,181)(158,182,160,184)(161,187,163,185)(162,186,164,188)(165,191,167,189)(166,190,168,192), (1,115,3,113)(2,24,4,22)(5,44,7,42)(6,133,8,135)(9,23,11,21)(10,116,12,114)(13,27,15,25)(14,120,16,118)(17,31,19,29)(18,124,20,122)(26,106,28,108)(30,110,32,112)(33,93,35,95)(34,188,36,186)(37,98,39,100)(38,192,40,190)(41,102,43,104)(45,83,47,81)(46,174,48,176)(49,87,51,85)(50,178,52,180)(53,91,55,89)(54,182,56,184)(57,71,59,69)(58,162,60,164)(61,75,63,73)(62,166,64,168)(65,79,67,77)(66,170,68,172)(70,152,72,150)(74,156,76,154)(78,160,80,158)(82,138,84,140)(86,142,88,144)(90,146,92,148)(94,128,96,126)(97,130,99,132)(101,134,103,136)(105,119,107,117)(109,123,111,121)(125,187,127,185)(129,191,131,189)(137,173,139,175)(141,177,143,179)(145,181,147,183)(149,163,151,161)(153,167,155,165)(157,171,159,169)>;
 
G:=Group( (1,109,105)(2,110,106)(3,111,107)(4,112,108)(5,192,188)(6,189,185)(7,190,186)(8,191,187)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(33,41,37)(34,42,38)(35,43,39)(36,44,40)(45,53,49)(46,54,50)(47,55,51)(48,56,52)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(69,77,73)(70,78,74)(71,79,75)(72,80,76)(81,89,85)(82,90,86)(83,91,87)(84,92,88)(93,102,98)(94,103,99)(95,104,100)(96,101,97)(113,121,117)(114,122,118)(115,123,119)(116,124,120)(125,133,129)(126,134,130)(127,135,131)(128,136,132)(137,145,141)(138,146,142)(139,147,143)(140,148,144)(149,157,153)(150,158,154)(151,159,155)(152,160,156)(161,169,165)(162,170,166)(163,171,167)(164,172,168)(173,181,177)(174,182,178)(175,183,179)(176,184,180), (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), (1,9)(2,10)(3,11)(4,12)(5,103)(6,104)(7,101)(8,102)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,151)(58,152)(59,149)(60,150)(61,155)(62,156)(63,153)(64,154)(65,159)(66,160)(67,157)(68,158)(69,163)(70,164)(71,161)(72,162)(73,167)(74,168)(75,165)(76,166)(77,171)(78,172)(79,169)(80,170)(81,175)(82,176)(83,173)(84,174)(85,179)(86,180)(87,177)(88,178)(89,183)(90,184)(91,181)(92,182)(93,187)(94,188)(95,185)(96,186)(97,190)(98,191)(99,192)(100,189), (1,149)(2,150)(3,151)(4,152)(5,148)(6,145)(7,146)(8,147)(9,59)(10,60)(11,57)(12,58)(13,63)(14,64)(15,61)(16,62)(17,67)(18,68)(19,65)(20,66)(21,71)(22,72)(23,69)(24,70)(25,75)(26,76)(27,73)(28,74)(29,79)(30,80)(31,77)(32,78)(33,83)(34,84)(35,81)(36,82)(37,87)(38,88)(39,85)(40,86)(41,91)(42,92)(43,89)(44,90)(45,95)(46,96)(47,93)(48,94)(49,100)(50,97)(51,98)(52,99)(53,104)(54,101)(55,102)(56,103)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192), (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), (1,127,3,125)(2,126,4,128)(5,170,7,172)(6,169,8,171)(9,35,11,33)(10,34,12,36)(13,39,15,37)(14,38,16,40)(17,43,19,41)(18,42,20,44)(21,47,23,45)(22,46,24,48)(25,51,27,49)(26,50,28,52)(29,55,31,53)(30,54,32,56)(57,83,59,81)(58,82,60,84)(61,87,63,85)(62,86,64,88)(65,91,67,89)(66,90,68,92)(69,95,71,93)(70,94,72,96)(73,100,75,98)(74,99,76,97)(77,104,79,102)(78,103,80,101)(105,131,107,129)(106,130,108,132)(109,135,111,133)(110,134,112,136)(113,139,115,137)(114,138,116,140)(117,143,119,141)(118,142,120,144)(121,147,123,145)(122,146,124,148)(149,175,151,173)(150,174,152,176)(153,179,155,177)(154,178,156,180)(157,183,159,181)(158,182,160,184)(161,187,163,185)(162,186,164,188)(165,191,167,189)(166,190,168,192), (1,115,3,113)(2,24,4,22)(5,44,7,42)(6,133,8,135)(9,23,11,21)(10,116,12,114)(13,27,15,25)(14,120,16,118)(17,31,19,29)(18,124,20,122)(26,106,28,108)(30,110,32,112)(33,93,35,95)(34,188,36,186)(37,98,39,100)(38,192,40,190)(41,102,43,104)(45,83,47,81)(46,174,48,176)(49,87,51,85)(50,178,52,180)(53,91,55,89)(54,182,56,184)(57,71,59,69)(58,162,60,164)(61,75,63,73)(62,166,64,168)(65,79,67,77)(66,170,68,172)(70,152,72,150)(74,156,76,154)(78,160,80,158)(82,138,84,140)(86,142,88,144)(90,146,92,148)(94,128,96,126)(97,130,99,132)(101,134,103,136)(105,119,107,117)(109,123,111,121)(125,187,127,185)(129,191,131,189)(137,173,139,175)(141,177,143,179)(145,181,147,183)(149,163,151,161)(153,167,155,165)(157,171,159,169) );
 
G=PermutationGroup([[(1,109,105),(2,110,106),(3,111,107),(4,112,108),(5,192,188),(6,189,185),(7,190,186),(8,191,187),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(21,29,25),(22,30,26),(23,31,27),(24,32,28),(33,41,37),(34,42,38),(35,43,39),(36,44,40),(45,53,49),(46,54,50),(47,55,51),(48,56,52),(57,65,61),(58,66,62),(59,67,63),(60,68,64),(69,77,73),(70,78,74),(71,79,75),(72,80,76),(81,89,85),(82,90,86),(83,91,87),(84,92,88),(93,102,98),(94,103,99),(95,104,100),(96,101,97),(113,121,117),(114,122,118),(115,123,119),(116,124,120),(125,133,129),(126,134,130),(127,135,131),(128,136,132),(137,145,141),(138,146,142),(139,147,143),(140,148,144),(149,157,153),(150,158,154),(151,159,155),(152,160,156),(161,169,165),(162,170,166),(163,171,167),(164,172,168),(173,181,177),(174,182,178),(175,183,179),(176,184,180)], [(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)], [(1,9),(2,10),(3,11),(4,12),(5,103),(6,104),(7,101),(8,102),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,151),(58,152),(59,149),(60,150),(61,155),(62,156),(63,153),(64,154),(65,159),(66,160),(67,157),(68,158),(69,163),(70,164),(71,161),(72,162),(73,167),(74,168),(75,165),(76,166),(77,171),(78,172),(79,169),(80,170),(81,175),(82,176),(83,173),(84,174),(85,179),(86,180),(87,177),(88,178),(89,183),(90,184),(91,181),(92,182),(93,187),(94,188),(95,185),(96,186),(97,190),(98,191),(99,192),(100,189)], [(1,149),(2,150),(3,151),(4,152),(5,148),(6,145),(7,146),(8,147),(9,59),(10,60),(11,57),(12,58),(13,63),(14,64),(15,61),(16,62),(17,67),(18,68),(19,65),(20,66),(21,71),(22,72),(23,69),(24,70),(25,75),(26,76),(27,73),(28,74),(29,79),(30,80),(31,77),(32,78),(33,83),(34,84),(35,81),(36,82),(37,87),(38,88),(39,85),(40,86),(41,91),(42,92),(43,89),(44,90),(45,95),(46,96),(47,93),(48,94),(49,100),(50,97),(51,98),(52,99),(53,104),(54,101),(55,102),(56,103),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,169),(122,170),(123,171),(124,172),(125,173),(126,174),(127,175),(128,176),(129,177),(130,178),(131,179),(132,180),(133,181),(134,182),(135,183),(136,184),(137,185),(138,186),(139,187),(140,188),(141,189),(142,190),(143,191),(144,192)], [(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)], [(1,127,3,125),(2,126,4,128),(5,170,7,172),(6,169,8,171),(9,35,11,33),(10,34,12,36),(13,39,15,37),(14,38,16,40),(17,43,19,41),(18,42,20,44),(21,47,23,45),(22,46,24,48),(25,51,27,49),(26,50,28,52),(29,55,31,53),(30,54,32,56),(57,83,59,81),(58,82,60,84),(61,87,63,85),(62,86,64,88),(65,91,67,89),(66,90,68,92),(69,95,71,93),(70,94,72,96),(73,100,75,98),(74,99,76,97),(77,104,79,102),(78,103,80,101),(105,131,107,129),(106,130,108,132),(109,135,111,133),(110,134,112,136),(113,139,115,137),(114,138,116,140),(117,143,119,141),(118,142,120,144),(121,147,123,145),(122,146,124,148),(149,175,151,173),(150,174,152,176),(153,179,155,177),(154,178,156,180),(157,183,159,181),(158,182,160,184),(161,187,163,185),(162,186,164,188),(165,191,167,189),(166,190,168,192)], [(1,115,3,113),(2,24,4,22),(5,44,7,42),(6,133,8,135),(9,23,11,21),(10,116,12,114),(13,27,15,25),(14,120,16,118),(17,31,19,29),(18,124,20,122),(26,106,28,108),(30,110,32,112),(33,93,35,95),(34,188,36,186),(37,98,39,100),(38,192,40,190),(41,102,43,104),(45,83,47,81),(46,174,48,176),(49,87,51,85),(50,178,52,180),(53,91,55,89),(54,182,56,184),(57,71,59,69),(58,162,60,164),(61,75,63,73),(62,166,64,168),(65,79,67,77),(66,170,68,172),(70,152,72,150),(74,156,76,154),(78,160,80,158),(82,138,84,140),(86,142,88,144),(90,146,92,148),(94,128,96,126),(97,130,99,132),(101,134,103,136),(105,119,107,117),(109,123,111,121),(125,187,127,185),(129,191,131,189),(137,173,139,175),(141,177,143,179),(145,181,147,183),(149,163,151,161),(153,167,155,165),(157,171,159,169)]])
 

66 conjugacy classes

class 1 2A···2G3A3B4A···4N6A···6N12A···12AB
order12···2334···46···612···12
size11···1114···41···14···4

66 irreducible representations

dim11111111222222
type+++++-
imageC1C2C2C2C3C6C6C6D4Q8C4○D4C3×D4C3×Q8C3×C4○D4
kernelC3×C23.78C23C3×C2.C42C6×C4⋊C4Q8×C2×C6C23.78C23C2.C42C2×C4⋊C4C22×Q8C2×C12C2×C12C2×C6C2×C4C2×C4C22
# reps1331266266212124

Matrix representation of C3×C23.78C23 ►in GL6(𝔽13)

900000
090000
001000
000100
000030
000003
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
0000120
0000012
,
1110000
1120000
0011700
003200
000001
0000120
,
500000
580000
003800
0021000
0000104
000043
,
500000
050000
0051100
000800
000001
0000120

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,11,3,0,0,0,0,7,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[5,5,0,0,0,0,0,8,0,0,0,0,0,0,3,2,0,0,0,0,8,10,0,0,0,0,0,0,10,4,0,0,0,0,4,3],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,11,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
 

C3×C23.78C23 in GAP, Magma, Sage, TeX 

C_3\times C_2^3._{78}C_2^3
 
% in TeX
 
G:=Group("C3xC2^3.78C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,828);
 
// by ID
 
G=gap.SmallGroup(192,828);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,176,1094,1059,142]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=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
 

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