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## G = Dic6.4Q8order 192 = 26·3

### 2nd non-split extension by Dic6 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6.4Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C4×Dic6 — Dic6.4Q8
 Lower central C3 — C6 — C2×C12 — Dic6.4Q8
 Upper central C1 — C22 — C42 — C42.C2

Generators and relations for Dic6.4Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=dad-1=a-1, cac-1=a7, cbc-1=a9b, bd=db, dcd-1=a3c-1 >

Subgroups: 208 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×7], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×3], Dic3 [×3], C12 [×2], C12 [×4], C2×C6, C42, C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, Q8.Q8, C12⋊C8, C6.Q16, C12.Q8, C6.SD16 [×2], C4×Dic6, C3×C42.C2, Dic6.4Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C4○D8, C8.C22, S3×Q8, Q83S3, C2×C3⋊D4, Q8.Q8, D63Q8, Q8.13D6, Q8.14D6, Dic6.4Q8

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

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

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

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

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 ··· 12F 12G 12H 12I 12J 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 size 1 1 1 1 2 2 2 2 2 4 8 8 12 12 12 12 2 2 2 12 12 12 12 4 ··· 4 8 8 8 8

33 irreducible representations

 dim 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 S3 Q8 D4 D6 D6 C4○D4 C3⋊D4 C4○D8 C8.C22 S3×Q8 Q8⋊3S3 Q8.13D6 Q8.14D6 kernel Dic6.4Q8 C12⋊C8 C6.Q16 C12.Q8 C6.SD16 C4×Dic6 C3×C42.C2 C42.C2 Dic6 C2×C12 C42 C4⋊C4 C12 C2×C4 C6 C6 C4 C4 C2 C2 # reps 1 1 1 1 2 1 1 1 2 2 1 2 2 4 4 1 1 1 2 2

Matrix representation of Dic6.4Q8 in GL6(𝔽73)

 1 48 0 0 0 0 3 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 69 38 0 0 0 0 13 4 0 0 0 0 0 0 18 5 0 0 0 0 23 55 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 21 57 0 0 0 0 64 52 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 38 4 0 0 0 0 59 35 0 0 0 0 0 0 18 5 0 0 0 0 23 55 0 0 0 0 0 0 12 1 0 0 0 0 1 61

G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[69,13,0,0,0,0,38,4,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[21,64,0,0,0,0,57,52,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[38,59,0,0,0,0,4,35,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,12,1,0,0,0,0,1,61] >;

Dic6.4Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6._4Q_8
% in TeX

G:=Group("Dic6.4Q8");
// GroupNames label

G:=SmallGroup(192,622);
// by ID

G=gap.SmallGroup(192,622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations

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