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## G = C8×Dic6order 192 = 26·3

### Direct product of C8 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C8×Dic6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C4×Dic6 — C8×Dic6
 Lower central C3 — C6 — C8×Dic6
 Upper central C1 — C2×C8 — C4×C8

Generators and relations for C8×Dic6
G = < a,b,c | a8=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 184 in 102 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, C24, Dic6, C2×Dic3, C2×C12, C4×C8, C4×C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8×Q8, C12⋊C8, C8×Dic3, Dic3⋊C8, C4×C24, C4×Dic6, C8×Dic6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Q8, C23, D6, C2×C8, C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C4×Q8, C22×C8, C8○D4, S3×C8, C2×Dic6, S3×C2×C4, C4○D12, C8×Q8, C4×Dic6, S3×C2×C8, C8○D12, C8×Dic6

Smallest permutation representation of C8×Dic6
Regular action on 192 points
Generators in S192
(1 15 59 151 75 124 106 181)(2 16 60 152 76 125 107 182)(3 17 49 153 77 126 108 183)(4 18 50 154 78 127 97 184)(5 19 51 155 79 128 98 185)(6 20 52 156 80 129 99 186)(7 21 53 145 81 130 100 187)(8 22 54 146 82 131 101 188)(9 23 55 147 83 132 102 189)(10 24 56 148 84 121 103 190)(11 13 57 149 73 122 104 191)(12 14 58 150 74 123 105 192)(25 89 119 38 69 138 166 172)(26 90 120 39 70 139 167 173)(27 91 109 40 71 140 168 174)(28 92 110 41 72 141 157 175)(29 93 111 42 61 142 158 176)(30 94 112 43 62 143 159 177)(31 95 113 44 63 144 160 178)(32 96 114 45 64 133 161 179)(33 85 115 46 65 134 162 180)(34 86 116 47 66 135 163 169)(35 87 117 48 67 136 164 170)(36 88 118 37 68 137 165 171)
(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)
(1 168 7 162)(2 167 8 161)(3 166 9 160)(4 165 10 159)(5 164 11 158)(6 163 12 157)(13 176 19 170)(14 175 20 169)(15 174 21 180)(16 173 22 179)(17 172 23 178)(18 171 24 177)(25 55 31 49)(26 54 32 60)(27 53 33 59)(28 52 34 58)(29 51 35 57)(30 50 36 56)(37 121 43 127)(38 132 44 126)(39 131 45 125)(40 130 46 124)(41 129 47 123)(42 128 48 122)(61 98 67 104)(62 97 68 103)(63 108 69 102)(64 107 70 101)(65 106 71 100)(66 105 72 99)(73 111 79 117)(74 110 80 116)(75 109 81 115)(76 120 82 114)(77 119 83 113)(78 118 84 112)(85 151 91 145)(86 150 92 156)(87 149 93 155)(88 148 94 154)(89 147 95 153)(90 146 96 152)(133 182 139 188)(134 181 140 187)(135 192 141 186)(136 191 142 185)(137 190 143 184)(138 189 144 183)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A 6B 6C 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 12A ··· 12L 24A ··· 24P order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 8 ··· 8 8 8 8 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 1 1 1 1 2 2 2 2 6 ··· 6 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 S3 Q8 D6 D6 C4○D4 Dic6 C4×S3 C8○D4 S3×C8 C4○D12 C8○D12 kernel C8×Dic6 C12⋊C8 C8×Dic3 Dic3⋊C8 C4×C24 C4×Dic6 Dic3⋊C4 C4⋊Dic3 C2×Dic6 Dic6 C4×C8 C24 C42 C2×C8 C12 C8 C2×C4 C6 C4 C4 C2 # reps 1 1 2 2 1 1 4 2 2 16 1 2 1 2 2 4 4 4 8 4 8

Matrix representation of C8×Dic6 in GL3(𝔽73) generated by

 22 0 0 0 1 0 0 0 1
,
 1 0 0 0 59 66 0 7 66
,
 1 0 0 0 10 22 0 12 63
G:=sub<GL(3,GF(73))| [22,0,0,0,1,0,0,0,1],[1,0,0,0,59,7,0,66,66],[1,0,0,0,10,12,0,22,63] >;

C8×Dic6 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_6
% in TeX

G:=Group("C8xDic6");
// GroupNames label

G:=SmallGroup(192,237);
// by ID

G=gap.SmallGroup(192,237);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,58,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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