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## G = Dic6⋊10Q8order 192 = 26·3

### The semidirect product of Dic6 and Q8 acting through Inn(Dic6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊10Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C2×Dic6 — C4×Dic6 — Dic6⋊10Q8
 Lower central C3 — C2×C6 — Dic6⋊10Q8
 Upper central C1 — C22 — C4×Q8

Generators and relations for Dic610Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd-1=c-1 >

Subgroups: 392 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C6×Q8, Q83Q8, C4×Dic6, C4×Dic6, C122Q8, C12.6Q8, Dic6⋊C4, Dic3.Q8, Dic3⋊Q8, Q8×C12, Dic610Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, 2- 1+4, C4○D12, S3×Q8, S3×C23, Q83Q8, C2×C4○D12, C2×S3×Q8, Q8○D12, Dic610Q8

Smallest permutation representation of Dic610Q8
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 70 7 64)(2 69 8 63)(3 68 9 62)(4 67 10 61)(5 66 11 72)(6 65 12 71)(13 106 19 100)(14 105 20 99)(15 104 21 98)(16 103 22 97)(17 102 23 108)(18 101 24 107)(25 81 31 75)(26 80 32 74)(27 79 33 73)(28 78 34 84)(29 77 35 83)(30 76 36 82)(37 91 43 85)(38 90 44 96)(39 89 45 95)(40 88 46 94)(41 87 47 93)(42 86 48 92)(49 151 55 145)(50 150 56 156)(51 149 57 155)(52 148 58 154)(53 147 59 153)(54 146 60 152)(109 181 115 187)(110 192 116 186)(111 191 117 185)(112 190 118 184)(113 189 119 183)(114 188 120 182)(121 134 127 140)(122 133 128 139)(123 144 129 138)(124 143 130 137)(125 142 131 136)(126 141 132 135)(157 172 163 178)(158 171 164 177)(159 170 165 176)(160 169 166 175)(161 180 167 174)(162 179 168 173)
(1 45 151 101)(2 46 152 102)(3 47 153 103)(4 48 154 104)(5 37 155 105)(6 38 156 106)(7 39 145 107)(8 40 146 108)(9 41 147 97)(10 42 148 98)(11 43 149 99)(12 44 150 100)(13 65 96 50)(14 66 85 51)(15 67 86 52)(16 68 87 53)(17 69 88 54)(18 70 89 55)(19 71 90 56)(20 72 91 57)(21 61 92 58)(22 62 93 59)(23 63 94 60)(24 64 95 49)(25 121 164 191)(26 122 165 192)(27 123 166 181)(28 124 167 182)(29 125 168 183)(30 126 157 184)(31 127 158 185)(32 128 159 186)(33 129 160 187)(34 130 161 188)(35 131 162 189)(36 132 163 190)(73 144 169 115)(74 133 170 116)(75 134 171 117)(76 135 172 118)(77 136 173 119)(78 137 174 120)(79 138 175 109)(80 139 176 110)(81 140 177 111)(82 141 178 112)(83 142 179 113)(84 143 180 114)
(1 30 151 157)(2 31 152 158)(3 32 153 159)(4 33 154 160)(5 34 155 161)(6 35 156 162)(7 36 145 163)(8 25 146 164)(9 26 147 165)(10 27 148 166)(11 28 149 167)(12 29 150 168)(13 142 96 113)(14 143 85 114)(15 144 86 115)(16 133 87 116)(17 134 88 117)(18 135 89 118)(19 136 90 119)(20 137 91 120)(21 138 92 109)(22 139 93 110)(23 140 94 111)(24 141 95 112)(37 188 105 130)(38 189 106 131)(39 190 107 132)(40 191 108 121)(41 192 97 122)(42 181 98 123)(43 182 99 124)(44 183 100 125)(45 184 101 126)(46 185 102 127)(47 186 103 128)(48 187 104 129)(49 178 64 82)(50 179 65 83)(51 180 66 84)(52 169 67 73)(53 170 68 74)(54 171 69 75)(55 172 70 76)(56 173 71 77)(57 174 72 78)(58 175 61 79)(59 176 62 80)(60 177 63 81)

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

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

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

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 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 C4○D4 C4○D12 2- 1+4 S3×Q8 Q8○D12 kernel Dic6⋊10Q8 C4×Dic6 C12⋊2Q8 C12.6Q8 Dic6⋊C4 Dic3.Q8 Dic3⋊Q8 Q8×C12 C4×Q8 Dic6 C42 C4⋊C4 C2×Q8 C12 C4 C6 C4 C2 # reps 1 3 1 2 2 4 2 1 1 4 3 3 1 4 8 1 2 2

Matrix representation of Dic610Q8 in GL4(𝔽13) generated by

 7 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1
,
 0 2 0 0 6 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0
,
 1 0 0 0 0 1 0 0 0 0 10 4 0 0 4 3
G:=sub<GL(4,GF(13))| [7,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,10,4,0,0,4,3] >;

Dic610Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{10}Q_8
% in TeX

G:=Group("Dic6:10Q8");
// GroupNames label

G:=SmallGroup(192,1126);
// by ID

G=gap.SmallGroup(192,1126);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,185,192,6278]);
// 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,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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