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## G = Q8×Dic6order 192 = 26·3

### Direct product of Q8 and Dic6

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 456 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C4×Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C2×Dic6, C6×Q8, Q82, C4×Dic6, C122Q8, C12⋊Q8, Q8×Dic3, Q8×C12, Q8×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, 2+ 1+4, C2×Dic6, S3×Q8, S3×C23, Q82, C22×Dic6, C2×S3×Q8, D4○D12, Q8×Dic6

Smallest permutation representation of Q8×Dic6
Regular action on 192 points
Generators in S192
(1 86 23 103)(2 87 24 104)(3 88 13 105)(4 89 14 106)(5 90 15 107)(6 91 16 108)(7 92 17 97)(8 93 18 98)(9 94 19 99)(10 95 20 100)(11 96 21 101)(12 85 22 102)(25 113 152 184)(26 114 153 185)(27 115 154 186)(28 116 155 187)(29 117 156 188)(30 118 145 189)(31 119 146 190)(32 120 147 191)(33 109 148 192)(34 110 149 181)(35 111 150 182)(36 112 151 183)(37 170 121 62)(38 171 122 63)(39 172 123 64)(40 173 124 65)(41 174 125 66)(42 175 126 67)(43 176 127 68)(44 177 128 69)(45 178 129 70)(46 179 130 71)(47 180 131 72)(48 169 132 61)(49 135 74 168)(50 136 75 157)(51 137 76 158)(52 138 77 159)(53 139 78 160)(54 140 79 161)(55 141 80 162)(56 142 81 163)(57 143 82 164)(58 144 83 165)(59 133 84 166)(60 134 73 167)
(1 39 23 123)(2 40 24 124)(3 41 13 125)(4 42 14 126)(5 43 15 127)(6 44 16 128)(7 45 17 129)(8 46 18 130)(9 47 19 131)(10 48 20 132)(11 37 21 121)(12 38 22 122)(25 75 152 50)(26 76 153 51)(27 77 154 52)(28 78 155 53)(29 79 156 54)(30 80 145 55)(31 81 146 56)(32 82 147 57)(33 83 148 58)(34 84 149 59)(35 73 150 60)(36 74 151 49)(61 100 169 95)(62 101 170 96)(63 102 171 85)(64 103 172 86)(65 104 173 87)(66 105 174 88)(67 106 175 89)(68 107 176 90)(69 108 177 91)(70 97 178 92)(71 98 179 93)(72 99 180 94)(109 144 192 165)(110 133 181 166)(111 134 182 167)(112 135 183 168)(113 136 184 157)(114 137 185 158)(115 138 186 159)(116 139 187 160)(117 140 188 161)(118 141 189 162)(119 142 190 163)(120 143 191 164)
(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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 57 19 51)(14 56 20 50)(15 55 21 49)(16 54 22 60)(17 53 23 59)(18 52 24 58)(25 126 31 132)(26 125 32 131)(27 124 33 130)(28 123 34 129)(29 122 35 128)(30 121 36 127)(37 151 43 145)(38 150 44 156)(39 149 45 155)(40 148 46 154)(41 147 47 153)(42 146 48 152)(61 113 67 119)(62 112 68 118)(63 111 69 117)(64 110 70 116)(65 109 71 115)(66 120 72 114)(85 167 91 161)(86 166 92 160)(87 165 93 159)(88 164 94 158)(89 163 95 157)(90 162 96 168)(97 139 103 133)(98 138 104 144)(99 137 105 143)(100 136 106 142)(101 135 107 141)(102 134 108 140)(169 184 175 190)(170 183 176 189)(171 182 177 188)(172 181 178 187)(173 192 179 186)(174 191 180 185)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P ··· 4U 6A 6B 6C 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 ··· 2 4 4 4 6 6 6 6 12 ··· 12 2 2 2 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - - + + + - + - + image C1 C2 C2 C2 C2 C2 S3 Q8 Q8 D6 D6 D6 Dic6 2+ 1+4 S3×Q8 D4○D12 kernel Q8×Dic6 C4×Dic6 C12⋊2Q8 C12⋊Q8 Q8×Dic3 Q8×C12 C4×Q8 Dic6 C3×Q8 C42 C4⋊C4 C2×Q8 Q8 C6 C4 C2 # reps 1 3 3 6 2 1 1 4 4 3 3 1 8 1 2 2

Matrix representation of Q8×Dic6 in GL4(𝔽13) generated by

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

Q8×Dic6 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_6
% in TeX

G:=Group("Q8xDic6");
// GroupNames label

G:=SmallGroup(192,1125);
// by ID

G=gap.SmallGroup(192,1125);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,675,80,6278]);
// Polycyclic

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

׿
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