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## G = (C4×Dic3)⋊8C4order 192 = 26·3

### 4th 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)⋊8C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — (C4×Dic3)⋊8C4
 Lower central C3 — C2×C6 — (C4×Dic3)⋊8C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C4×Dic3)⋊8C4
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=b3c >

Subgroups: 408 in 186 conjugacy classes, 95 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×12], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×24], C23, Dic3 [×8], Dic3 [×2], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C42 [×4], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×12], C2×Dic3 [×6], C2×C12 [×6], C2×C12 [×6], C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C429C4, C2×C4×Dic3, C2×Dic3⋊C4 [×4], C2×C4⋊Dic3, C6×C4⋊C4, (C4×Dic3)⋊8C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×6], C23, D6 [×3], C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, S3×D4 [×2], S3×Q8 [×2], C2×C3⋊D4, C429C4, C12⋊Q8 [×2], S3×C4⋊C4 [×2], C2×Dic3⋊C4, C123D4, Dic3⋊Q8, (C4×Dic3)⋊8C4

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

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

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

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

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

Matrix representation of (C4×Dic3)⋊8C4 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 3 0 0 0 0 0 0 10 9 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 10 0 0 0 0 0 8 5 0 0 0 0 0 0 0 9 10 0 0 0 0 0 10 4
,
 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 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 0

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0,0,0,0,10,5,0,0,0,0,0,0,0,9,10,0,0,0,0,0,10,4],[8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0] >;

(C4×Dic3)⋊8C4 in GAP, Magma, Sage, TeX

(C_4\times {\rm Dic}_3)\rtimes_8C_4
% in TeX

G:=Group("(C4xDic3):8C4");
// GroupNames label

G:=SmallGroup(192,534);
// by ID

G=gap.SmallGroup(192,534);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,422,387,58,6278]);
// 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=b^3*c>;
// generators/relations

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