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## G = C3×C23.67C23order 192 = 26·3

### Direct product of C3 and C23.67C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23.67C23
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C2.C42 — C3×C23.67C23
 Lower central C1 — C22 — C3×C23.67C23
 Upper central C1 — C22×C6 — C3×C23.67C23

Generators and relations for C3×C23.67C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=d, f2=bcd, g2=c, 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, df=fd, dg=gd, fg=gf >

Subgroups: 274 in 186 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×C12, C2×C12, C3×Q8, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4×C12, C3×C4⋊C4, C22×C12, C22×C12, C6×Q8, C6×Q8, C23.67C23, C3×C2.C42, C2×C4×C12, C6×C4⋊C4, Q8×C2×C6, C3×C23.67C23
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C3×C22⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23.67C23, C6×C22⋊C4, Q8×C12, C3×C22⋊Q8, C3×C4.4D4, C3×C4⋊Q8, C3×C23.67C23

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

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 4M ··· 4T 6A ··· 6N 12A ··· 12X 12Y ··· 12AN order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C3×C23.67C23 C3×C2.C42 C2×C4×C12 C6×C4⋊C4 Q8×C2×C6 C23.67C23 C6×Q8 C2.C42 C2×C42 C2×C4⋊C4 C22×Q8 C2×Q8 C2×C12 C2×C12 C2×C6 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 2 8 8 2 2 2 16 4 4 4 8 8 8

Matrix representation of C3×C23.67C23 in GL5(𝔽13)

 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 7 6 0 0 0 9 6
,
 8 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 11 0 0 0 1 1

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

C3×C23.67C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{67}C_2^3
% in TeX

G:=Group("C3xC2^3.67C2^3");
// GroupNames label

G:=SmallGroup(192,824);
// by ID

G=gap.SmallGroup(192,824);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1184,1094,268]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=d,f^2=b*c*d,g^2=c,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,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations

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