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## G = C8.8Dic6order 192 = 26·3

### 5th non-split extension by C8 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8.8Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C8×Dic3 — C8.8Dic6
 Lower central C3 — C6 — C2×C12 — C8.8Dic6
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8.8Dic6
G = < a,b,c | a8=b12=1, c2=b6, bab-1=a3, ac=ca, cbc-1=a4b-1 >

Subgroups: 208 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×C8, C4.Q8, C4.Q8, C2.D8, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C8.5Q8, C6.Q16, C8×Dic3, C8⋊Dic3, C3×C4.Q8, C4.Dic6, C8.8Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C4○D8, C2×Dic6, S3×D4, S3×Q8, C8.5Q8, C12⋊Q8, Q8.7D6, C8.8Dic6

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - - + + + - - + image C1 C2 C2 C2 C2 C2 S3 Q8 Q8 D4 D6 D6 Dic6 C4○D8 S3×Q8 S3×D4 Q8.7D6 kernel C8.8Dic6 C6.Q16 C8×Dic3 C8⋊Dic3 C3×C4.Q8 C4.Dic6 C4.Q8 C3⋊C8 C24 C2×Dic3 C4⋊C4 C2×C8 C8 C6 C4 C22 C2 # reps 1 2 1 1 1 2 1 2 2 2 2 1 4 8 1 1 4

Matrix representation of C8.8Dic6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 18 0 0 69 61
,
 59 66 0 0 7 66 0 0 0 0 21 64 0 0 57 52
,
 47 8 0 0 34 26 0 0 0 0 27 8 0 0 55 46
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,69,0,0,18,61],[59,7,0,0,66,66,0,0,0,0,21,57,0,0,64,52],[47,34,0,0,8,26,0,0,0,0,27,55,0,0,8,46] >;

C8.8Dic6 in GAP, Magma, Sage, TeX

C_8._8{\rm Dic}_6
% in TeX

G:=Group("C8.8Dic6");
// GroupNames label

G:=SmallGroup(192,417);
// by ID

G=gap.SmallGroup(192,417);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,926,555,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^12=1,c^2=b^6,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^4*b^-1>;
// generators/relations

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