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## G = (C4×Dic3)⋊9C4order 192 = 26·3

### 5th semidirect product of C4×Dic3 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C4×Dic3)⋊9C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — (C4×Dic3)⋊9C4
 Lower central C3 — C2×C6 — (C4×Dic3)⋊9C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C4×Dic3)⋊9C4
G = < a,b,c,d | a4=b6=d4=1, c2=b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b3c >

Subgroups: 344 in 154 conjugacy classes, 79 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×8], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×20], C23, Dic3 [×6], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C42 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×6], C2×C12 [×6], C22×C6, C2.C42 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C428C4, C6.C42 [×4], C2×C4×Dic3, C2×C4⋊Dic3, C6×C4⋊C4, (C4×Dic3)⋊9C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, D42S3 [×2], Q83S3 [×2], C2×C3⋊D4, C428C4, C4.Dic6 [×2], C4⋊C47S3 [×2], C2×Dic3⋊C4, C23.12D6, C12.23D4, (C4×Dic3)⋊9C4

Smallest permutation representation of (C4×Dic3)⋊9C4
Regular action on 192 points
Generators in S192
(1 65 17 60)(2 66 18 55)(3 61 13 56)(4 62 14 57)(5 63 15 58)(6 64 16 59)(7 148 192 144)(8 149 187 139)(9 150 188 140)(10 145 189 141)(11 146 190 142)(12 147 191 143)(19 78 30 67)(20 73 25 68)(21 74 26 69)(22 75 27 70)(23 76 28 71)(24 77 29 72)(31 90 42 79)(32 85 37 80)(33 86 38 81)(34 87 39 82)(35 88 40 83)(36 89 41 84)(43 97 54 91)(44 98 49 92)(45 99 50 93)(46 100 51 94)(47 101 52 95)(48 102 53 96)(103 161 113 151)(104 162 114 152)(105 157 109 153)(106 158 110 154)(107 159 111 155)(108 160 112 156)(115 173 125 163)(116 174 126 164)(117 169 121 165)(118 170 122 166)(119 171 123 167)(120 172 124 168)(127 185 137 175)(128 186 138 176)(129 181 133 177)(130 182 134 178)(131 183 135 179)(132 184 136 180)
(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)
(1 121 4 124)(2 126 5 123)(3 125 6 122)(7 84 10 81)(8 83 11 80)(9 82 12 79)(13 115 16 118)(14 120 17 117)(15 119 18 116)(19 114 22 111)(20 113 23 110)(21 112 24 109)(25 103 28 106)(26 108 29 105)(27 107 30 104)(31 150 34 147)(32 149 35 146)(33 148 36 145)(37 139 40 142)(38 144 41 141)(39 143 42 140)(43 138 46 135)(44 137 47 134)(45 136 48 133)(49 127 52 130)(50 132 53 129)(51 131 54 128)(55 174 58 171)(56 173 59 170)(57 172 60 169)(61 163 64 166)(62 168 65 165)(63 167 66 164)(67 162 70 159)(68 161 71 158)(69 160 72 157)(73 151 76 154)(74 156 77 153)(75 155 78 152)(85 187 88 190)(86 192 89 189)(87 191 90 188)(91 186 94 183)(92 185 95 182)(93 184 96 181)(97 176 100 179)(98 175 101 178)(99 180 102 177)
(1 48 24 36)(2 43 19 31)(3 44 20 32)(4 45 21 33)(5 46 22 34)(6 47 23 35)(7 169 184 157)(8 170 185 158)(9 171 186 159)(10 172 181 160)(11 173 182 161)(12 174 183 162)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(55 97 67 90)(56 98 68 85)(57 99 69 86)(58 100 70 87)(59 101 71 88)(60 102 72 89)(61 92 73 80)(62 93 74 81)(63 94 75 82)(64 95 76 83)(65 96 77 84)(66 91 78 79)(103 146 115 134)(104 147 116 135)(105 148 117 136)(106 149 118 137)(107 150 119 138)(108 145 120 133)(109 144 121 132)(110 139 122 127)(111 140 123 128)(112 141 124 129)(113 142 125 130)(114 143 126 131)(151 190 163 178)(152 191 164 179)(153 192 165 180)(154 187 166 175)(155 188 167 176)(156 189 168 177)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 C4○D4 Dic6 C4×S3 C3⋊D4 D4⋊2S3 Q8⋊3S3 kernel (C4×Dic3)⋊9C4 C6.C42 C2×C4×Dic3 C2×C4⋊Dic3 C6×C4⋊C4 C4×Dic3 C2×C4⋊C4 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 4 1 1 1 8 1 2 2 3 8 4 4 4 2 2

Matrix representation of (C4×Dic3)⋊9C4 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 12 2 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 0 8 10 0 0 0 0 0 8 5 0 0 0 0 0 0 0 6 8 0 0 0 0 0 10 7 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 1 11 0 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,2,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,10,5,0,0,0,0,0,0,0,6,10,0,0,0,0,0,8,7,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[5,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,11,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

(C4×Dic3)⋊9C4 in GAP, Magma, Sage, TeX

(C_4\times {\rm Dic}_3)\rtimes_9C_4
% in TeX

G:=Group("(C4xDic3):9C4");
// GroupNames label

G:=SmallGroup(192,536);
// by ID

G=gap.SmallGroup(192,536);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,1094,387,58,6278]);
// Polycyclic

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

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