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## G = (C3×D12)⋊S3order 432 = 24·33

### 6th semidirect product of C3×D12 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — (C3×D12)⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — (C3×D12)⋊S3
 Lower central C33 — C32×C6 — (C3×D12)⋊S3
 Upper central C1 — C2 — C4

Generators and relations for (C3×D12)⋊S3
G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1464 in 288 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, D6⋊S3, S3×C12, C3×D12, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C3×C3⋊Dic3, C335C4, C32×C12, S3×C3×C6, C6×C3⋊S3, D125S3, C12.D6, S3×C3⋊Dic3, C336D4, C32×D12, C12×C3⋊S3, C338Q8, (C3×D12)⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, D42S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D125S3, C12.D6, C2×S3×C3⋊S3, (C3×D12)⋊S3

Smallest permutation representation of (C3×D12)⋊S3
On 144 points
Generators in S144
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)(97 105 101)(98 106 102)(99 107 103)(100 108 104)(109 117 113)(110 118 114)(111 119 115)(112 120 116)(121 129 125)(122 130 126)(123 131 127)(124 132 128)(133 137 141)(134 138 142)(135 139 143)(136 140 144)
(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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 108)(11 107)(12 106)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 96)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 84)(37 130)(38 129)(39 128)(40 127)(41 126)(42 125)(43 124)(44 123)(45 122)(46 121)(47 132)(48 131)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 120)(60 119)(61 142)(62 141)(63 140)(64 139)(65 138)(66 137)(67 136)(68 135)(69 134)(70 133)(71 144)(72 143)
(1 37 52)(2 38 53)(3 39 54)(4 40 55)(5 41 56)(6 42 57)(7 43 58)(8 44 59)(9 45 60)(10 46 49)(11 47 50)(12 48 51)(13 139 28)(14 140 29)(15 141 30)(16 142 31)(17 143 32)(18 144 33)(19 133 34)(20 134 35)(21 135 36)(22 136 25)(23 137 26)(24 138 27)(61 77 92)(62 78 93)(63 79 94)(64 80 95)(65 81 96)(66 82 85)(67 83 86)(68 84 87)(69 73 88)(70 74 89)(71 75 90)(72 76 91)(97 122 119)(98 123 120)(99 124 109)(100 125 110)(101 126 111)(102 127 112)(103 128 113)(104 129 114)(105 130 115)(106 131 116)(107 132 117)(108 121 118)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 61)(10 62)(11 63)(12 64)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 121)(22 122)(23 123)(24 124)(25 119)(26 120)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 73)(57 74)(58 75)(59 76)(60 77)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 133)(107 134)(108 135)

G:=sub<Sym(144)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,108)(11,107)(12,106)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,96)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,84)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,132)(48,131)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,120)(60,119)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,144)(72,143), (1,37,52)(2,38,53)(3,39,54)(4,40,55)(5,41,56)(6,42,57)(7,43,58)(8,44,59)(9,45,60)(10,46,49)(11,47,50)(12,48,51)(13,139,28)(14,140,29)(15,141,30)(16,142,31)(17,143,32)(18,144,33)(19,133,34)(20,134,35)(21,135,36)(22,136,25)(23,137,26)(24,138,27)(61,77,92)(62,78,93)(63,79,94)(64,80,95)(65,81,96)(66,82,85)(67,83,86)(68,84,87)(69,73,88)(70,74,89)(71,75,90)(72,76,91)(97,122,119)(98,123,120)(99,124,109)(100,125,110)(101,126,111)(102,127,112)(103,128,113)(104,129,114)(105,130,115)(106,131,116)(107,132,117)(108,121,118), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,61)(10,62)(11,63)(12,64)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,121)(22,122)(23,123)(24,124)(25,119)(26,120)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,133)(107,134)(108,135)>;

G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,101)(98,106,102)(99,107,103)(100,108,104)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,108)(11,107)(12,106)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,96)(25,83)(26,82)(27,81)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,84)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,132)(48,131)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,120)(60,119)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,144)(72,143), (1,37,52)(2,38,53)(3,39,54)(4,40,55)(5,41,56)(6,42,57)(7,43,58)(8,44,59)(9,45,60)(10,46,49)(11,47,50)(12,48,51)(13,139,28)(14,140,29)(15,141,30)(16,142,31)(17,143,32)(18,144,33)(19,133,34)(20,134,35)(21,135,36)(22,136,25)(23,137,26)(24,138,27)(61,77,92)(62,78,93)(63,79,94)(64,80,95)(65,81,96)(66,82,85)(67,83,86)(68,84,87)(69,73,88)(70,74,89)(71,75,90)(72,76,91)(97,122,119)(98,123,120)(99,124,109)(100,125,110)(101,126,111)(102,127,112)(103,128,113)(104,129,114)(105,130,115)(106,131,116)(107,132,117)(108,121,118), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,61)(10,62)(11,63)(12,64)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,121)(22,122)(23,123)(24,124)(25,119)(26,120)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,133)(107,134)(108,135) );

G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92),(97,105,101),(98,106,102),(99,107,103),(100,108,104),(109,117,113),(110,118,114),(111,119,115),(112,120,116),(121,129,125),(122,130,126),(123,131,127),(124,132,128),(133,137,141),(134,138,142),(135,139,143),(136,140,144)], [(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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,108),(11,107),(12,106),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,96),(25,83),(26,82),(27,81),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,84),(37,130),(38,129),(39,128),(40,127),(41,126),(42,125),(43,124),(44,123),(45,122),(46,121),(47,132),(48,131),(49,118),(50,117),(51,116),(52,115),(53,114),(54,113),(55,112),(56,111),(57,110),(58,109),(59,120),(60,119),(61,142),(62,141),(63,140),(64,139),(65,138),(66,137),(67,136),(68,135),(69,134),(70,133),(71,144),(72,143)], [(1,37,52),(2,38,53),(3,39,54),(4,40,55),(5,41,56),(6,42,57),(7,43,58),(8,44,59),(9,45,60),(10,46,49),(11,47,50),(12,48,51),(13,139,28),(14,140,29),(15,141,30),(16,142,31),(17,143,32),(18,144,33),(19,133,34),(20,134,35),(21,135,36),(22,136,25),(23,137,26),(24,138,27),(61,77,92),(62,78,93),(63,79,94),(64,80,95),(65,81,96),(66,82,85),(67,83,86),(68,84,87),(69,73,88),(70,74,89),(71,75,90),(72,76,91),(97,122,119),(98,123,120),(99,124,109),(100,125,110),(101,126,111),(102,127,112),(103,128,113),(104,129,114),(105,130,115),(106,131,116),(107,132,117),(108,121,118)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,61),(10,62),(11,63),(12,64),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,121),(22,122),(23,123),(24,124),(25,119),(26,120),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,96),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,73),(57,74),(58,75),(59,76),(60,77),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,133),(107,134),(108,135)]])

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 size 1 1 6 6 18 2 ··· 2 4 4 4 4 2 9 9 54 54 2 ··· 2 4 4 4 4 12 ··· 12 18 18 2 2 4 ··· 4 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 C2×S32 D12⋊5S3 kernel (C3×D12)⋊S3 S3×C3⋊Dic3 C33⋊6D4 C32×D12 C12×C3⋊S3 C33⋊8Q8 C3×D12 C4×C3⋊S3 C3⋊Dic3 C3×C12 S3×C6 C2×C3⋊S3 C33 C32 C12 C32 C6 C3 # reps 1 2 2 1 1 1 4 1 1 5 8 1 2 4 4 4 4 8

Matrix representation of (C3×D12)⋊S3 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 8 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 5 3 0 0 0 0 0 0 5 8 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,5,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;

(C3×D12)⋊S3 in GAP, Magma, Sage, TeX

(C_3\times D_{12})\rtimes S_3
% in TeX

G:=Group("(C3xD12):S3");
// GroupNames label

G:=SmallGroup(432,661);
// by ID

G=gap.SmallGroup(432,661);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,571,2028,14118]);
// Polycyclic

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

׿
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𝔽