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## G = Dic6.Q8order 192 = 26·3

### 1st non-split extension by Dic6 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic6.Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, dad-1=a5, cbc-1=a3b, bd=db, dcd-1=a6c-1 >

Subgroups: 224 in 90 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, Q8.Q8, C6.Q16, C6.SD16, Dic3⋊C8, C2.Dic12, C3×C4.Q8, Dic6⋊C4, C4.Dic6, Dic6.Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D8, C8.C22, C4○D12, S3×D4, S3×Q8, Q8.Q8, D6⋊Q8, D4.D6, Q8.7D6, Dic6.Q8

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 C4○D4 C4○D8 C4○D12 C8.C22 S3×Q8 S3×D4 D4.D6 Q8.7D6 kernel Dic6.Q8 C6.Q16 C6.SD16 Dic3⋊C8 C2.Dic12 C3×C4.Q8 Dic6⋊C4 C4.Dic6 C4.Q8 Dic6 C2×Dic3 C4⋊C4 C2×C8 C12 C6 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 2 4 4 1 1 1 2 2

Matrix representation of Dic6.Q8 in GL4(𝔽73) generated by

 1 70 0 0 25 72 0 0 0 0 1 72 0 0 1 0
,
 69 60 0 0 35 4 0 0 0 0 71 13 0 0 11 2
,
 17 49 0 0 6 56 0 0 0 0 7 59 0 0 14 66
,
 46 0 0 0 0 46 0 0 0 0 54 14 0 0 68 19
G:=sub<GL(4,GF(73))| [1,25,0,0,70,72,0,0,0,0,1,1,0,0,72,0],[69,35,0,0,60,4,0,0,0,0,71,11,0,0,13,2],[17,6,0,0,49,56,0,0,0,0,7,14,0,0,59,66],[46,0,0,0,0,46,0,0,0,0,54,68,0,0,14,19] >;

Dic6.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6.Q_8
% in TeX

G:=Group("Dic6.Q8");
// GroupNames label

G:=SmallGroup(192,416);
// by ID

G=gap.SmallGroup(192,416);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,1094,135,100,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,d*a*d^-1=a^5,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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