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## G = C8⋊Dic6order 192 = 26·3

### 1st semidirect product of C8 and Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8⋊Dic6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12⋊2Q8 — C8⋊Dic6
 Lower central C3 — C6 — C2×C12 — C8⋊Dic6
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C8⋊Dic6
G = < a,b,c | a8=b12=1, c2=b6, bab-1=a5, cac-1=a-1, cbc-1=b-1 >

Subgroups: 248 in 90 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊Q8, C8⋊Dic3, C241C4, C3×C8⋊C4, C122Q8, C12.6Q8, C8⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, D12, C22×S3, C4⋊Q8, C8⋊C22, C8.C22, C2×Dic6, C2×D12, C8⋊Q8, C122Q8, C8⋊D6, C8.D6, C8⋊Dic6

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 4 4 24 24 24 24 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - + + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 Dic6 D12 C8⋊C22 C8.C22 C8⋊D6 C8.D6 kernel C8⋊Dic6 C8⋊Dic3 C24⋊1C4 C3×C8⋊C4 C12⋊2Q8 C12.6Q8 C8⋊C4 C24 C2×C12 C42 C2×C8 C8 C2×C4 C6 C6 C2 C2 # reps 1 2 2 1 1 1 1 4 2 1 2 8 4 1 1 2 2

Matrix representation of C8⋊Dic6 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 22 11 49 48 0 0 0 0 30 5 38 8 0 0 0 0 8 24 51 62 0 0 0 0 19 24 43 68
,
 52 70 0 0 0 0 0 0 50 21 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 24 25 68 4 0 0 0 0 35 65 64 22 0 0 0 0 22 11 49 48 0 0 0 0 30 5 38 8
,
 40 70 0 0 0 0 0 0 47 33 0 0 0 0 0 0 0 0 11 60 0 0 0 0 0 0 71 62 0 0 0 0 0 0 0 0 8 37 39 44 0 0 0 0 46 65 52 34 0 0 0 0 34 29 16 61 0 0 0 0 21 39 8 57

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,30,8,19,0,0,0,0,11,5,24,24,0,0,0,0,49,38,51,43,0,0,0,0,48,8,62,68],[52,50,0,0,0,0,0,0,70,21,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,24,35,22,30,0,0,0,0,25,65,11,5,0,0,0,0,68,64,49,38,0,0,0,0,4,22,48,8],[40,47,0,0,0,0,0,0,70,33,0,0,0,0,0,0,0,0,11,71,0,0,0,0,0,0,60,62,0,0,0,0,0,0,0,0,8,46,34,21,0,0,0,0,37,65,29,39,0,0,0,0,39,52,16,8,0,0,0,0,44,34,61,57] >;

C8⋊Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("C8:Dic6");
// GroupNames label

G:=SmallGroup(192,261);
// by ID

G=gap.SmallGroup(192,261);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,387,58,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^12=1,c^2=b^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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