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G = Dic6.3Q8order 192 = 26·3

1st non-split extension by Dic6 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6.3Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C4×Dic6 — Dic6.3Q8
 Lower central C3 — C6 — C2×C12 — Dic6.3Q8
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for Dic6.3Q8
G = < a,b,c,d | a12=1, b2=c4=a6, d2=a3c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a3b, bd=db, dcd-1=c3 >

Subgroups: 232 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, Q8.Q8, C2.Dic12, C8⋊Dic3, C241C4, C3×C4⋊C8, C4×Dic6, C12.6Q8, Dic6.3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C4○D8, C8.C22, C2×D12, D42S3, S3×Q8, Q8.Q8, C4.D12, C4○D24, C8.D6, Dic6.3Q8

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

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

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

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

39 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 12G 12H 24A ··· 24H 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 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 4 12 12 12 12 24 24 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + + - - - - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 C4○D4 D12 C4○D8 C4○D24 C8.C22 D4⋊2S3 S3×Q8 C8.D6 kernel Dic6.3Q8 C2.Dic12 C8⋊Dic3 C24⋊1C4 C3×C4⋊C8 C4×Dic6 C12.6Q8 C4⋊C8 Dic6 C2×C12 C42 C2×C8 C12 C2×C4 C6 C2 C6 C4 C4 C2 # reps 1 2 1 1 1 1 1 1 2 2 1 2 2 4 4 8 1 1 1 2

Matrix representation of Dic6.3Q8 in GL6(𝔽73)

 27 10 0 0 0 0 0 46 0 0 0 0 0 0 65 0 0 0 0 0 55 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 42 26 0 0 0 0 36 31 0 0 0 0 0 0 46 62 0 0 0 0 33 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 60 0 0 0 0 0 51 0 0 0 0 0 0 27 0 0 0 0 0 40 46 0 0 0 0 0 0 63 3 0 0 0 0 15 10
,
 34 28 0 0 0 0 50 39 0 0 0 0 0 0 27 11 0 0 0 0 40 46 0 0 0 0 0 0 23 16 0 0 0 0 49 50

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,10,46,0,0,0,0,0,0,65,55,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,36,0,0,0,0,26,31,0,0,0,0,0,0,46,33,0,0,0,0,62,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,60,51,0,0,0,0,0,0,27,40,0,0,0,0,0,46,0,0,0,0,0,0,63,15,0,0,0,0,3,10],[34,50,0,0,0,0,28,39,0,0,0,0,0,0,27,40,0,0,0,0,11,46,0,0,0,0,0,0,23,49,0,0,0,0,16,50] >;

Dic6.3Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6._3Q_8
% in TeX

G:=Group("Dic6.3Q8");
// GroupNames label

G:=SmallGroup(192,388);
// by ID

G=gap.SmallGroup(192,388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,58,1123,136,6278]);
// Polycyclic

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

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