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## G = (C2×Dic6)⋊7C4order 192 = 26·3

### 3rd semidirect product of C2×Dic6 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×Dic6)⋊7C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C22×Dic6 — (C2×Dic6)⋊7C4
 Lower central C3 — C2×C6 — (C2×Dic6)⋊7C4
 Upper central C1 — C23 — C2×C42

Generators and relations for (C2×Dic6)⋊7C4
G = < a,b,c,d | a2=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=ab6c >

Subgroups: 440 in 186 conjugacy classes, 87 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×18], Q8 [×8], C23, Dic3 [×6], C12 [×4], C12 [×4], C2×C6 [×3], C2×C6 [×4], C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×8], Dic6 [×8], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×10], C2×C12 [×4], C22×C6, C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C4⋊Dic3 [×2], C4×C12 [×2], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C23.67C23, C6.C42 [×4], C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, (C2×Dic6)⋊7C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], Dic6 [×4], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, D6⋊C4 [×4], C2×Dic6 [×2], S3×C2×C4, C2×D12, C4○D12 [×2], C2×C3⋊D4, C23.67C23, C4×Dic6 [×2], C122Q8, C427S3, C12.48D4 [×2], C2×D6⋊C4, (C2×Dic6)⋊7C4

Smallest permutation representation of (C2×Dic6)⋊7C4
Regular action on 192 points
Generators in S192
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 97)(23 98)(24 99)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 145)(109 176)(110 177)(111 178)(112 179)(113 180)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 188)(122 189)(123 190)(124 191)(125 192)(126 181)(127 182)(128 183)(129 184)(130 185)(131 186)(132 187)(133 164)(134 165)(135 166)(136 167)(137 168)(138 157)(139 158)(140 159)(141 160)(142 161)(143 162)(144 163)
(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)
(1 118 7 112)(2 117 8 111)(3 116 9 110)(4 115 10 109)(5 114 11 120)(6 113 12 119)(13 148 19 154)(14 147 20 153)(15 146 21 152)(16 145 22 151)(17 156 23 150)(18 155 24 149)(25 132 31 126)(26 131 32 125)(27 130 33 124)(28 129 34 123)(29 128 35 122)(30 127 36 121)(37 140 43 134)(38 139 44 133)(39 138 45 144)(40 137 46 143)(41 136 47 142)(42 135 48 141)(49 108 55 102)(50 107 56 101)(51 106 57 100)(52 105 58 99)(53 104 59 98)(54 103 60 97)(61 165 67 159)(62 164 68 158)(63 163 69 157)(64 162 70 168)(65 161 71 167)(66 160 72 166)(73 173 79 179)(74 172 80 178)(75 171 81 177)(76 170 82 176)(77 169 83 175)(78 180 84 174)(85 185 91 191)(86 184 92 190)(87 183 93 189)(88 182 94 188)(89 181 95 187)(90 192 96 186)
(1 15 65 27)(2 16 66 28)(3 17 67 29)(4 18 68 30)(5 19 69 31)(6 20 70 32)(7 21 71 33)(8 22 72 34)(9 23 61 35)(10 24 62 36)(11 13 63 25)(12 14 64 26)(37 87 75 104)(38 88 76 105)(39 89 77 106)(40 90 78 107)(41 91 79 108)(42 92 80 97)(43 93 81 98)(44 94 82 99)(45 95 83 100)(46 96 84 101)(47 85 73 102)(48 86 74 103)(49 167 185 112)(50 168 186 113)(51 157 187 114)(52 158 188 115)(53 159 189 116)(54 160 190 117)(55 161 191 118)(56 162 192 119)(57 163 181 120)(58 164 182 109)(59 165 183 110)(60 166 184 111)(121 170 149 139)(122 171 150 140)(123 172 151 141)(124 173 152 142)(125 174 153 143)(126 175 154 144)(127 176 155 133)(128 177 156 134)(129 178 145 135)(130 179 146 136)(131 180 147 137)(132 169 148 138)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4L 4M ··· 4T 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 C4○D4 Dic6 C4×S3 D12 C3⋊D4 C4○D12 kernel (C2×Dic6)⋊7C4 C6.C42 C2×C4⋊Dic3 C2×C4×C12 C22×Dic6 C2×Dic6 C2×C42 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 8 1 4 4 3 4 8 4 4 4 8

Matrix representation of (C2×Dic6)⋊7C4 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 3 10 0 0 0 3 6 0 0 0 0 0 8 10 0 0 0 0 5
,
 1 0 0 0 0 0 5 5 0 0 0 0 8 0 0 0 0 0 11 9 0 0 0 11 2
,
 8 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 3 0 0 0 0 8

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

(C2×Dic6)⋊7C4 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_6)\rtimes_7C_4
% in TeX

G:=Group("(C2xDic6):7C4");
// GroupNames label

G:=SmallGroup(192,488);
// by ID

G=gap.SmallGroup(192,488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,758,58,6278]);
// Polycyclic

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

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