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## G = C3⋊S3×Dic6order 432 = 24·33

### Direct product of C3⋊S3 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C3⋊S3×Dic6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — Dic3×C3⋊S3 — C3⋊S3×Dic6
 Lower central C33 — C32×C6 — C3⋊S3×Dic6
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3×Dic6
G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=d6, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1368 in 276 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×Dic6, S3×Q8, C3×C3⋊S3, C32×C6, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C335C4, C32×C12, C6×C3⋊S3, S3×Dic6, Q8×C3⋊S3, Dic3×C3⋊S3, C334Q8, C32×Dic6, C12×C3⋊S3, C338Q8, C3⋊S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, S32, C2×C3⋊S3, C2×Dic6, S3×Q8, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×Dic6, Q8×C3⋊S3, C2×S3×C3⋊S3, C3⋊S3×Dic6

Smallest permutation representation of C3⋊S3×Dic6
On 144 points
Generators in S144
(1 20 138)(2 21 139)(3 22 140)(4 23 141)(5 24 142)(6 13 143)(7 14 144)(8 15 133)(9 16 134)(10 17 135)(11 18 136)(12 19 137)(25 109 82)(26 110 83)(27 111 84)(28 112 73)(29 113 74)(30 114 75)(31 115 76)(32 116 77)(33 117 78)(34 118 79)(35 119 80)(36 120 81)(37 49 96)(38 50 85)(39 51 86)(40 52 87)(41 53 88)(42 54 89)(43 55 90)(44 56 91)(45 57 92)(46 58 93)(47 59 94)(48 60 95)(61 101 128)(62 102 129)(63 103 130)(64 104 131)(65 105 132)(66 106 121)(67 107 122)(68 108 123)(69 97 124)(70 98 125)(71 99 126)(72 100 127)
(1 142 16)(2 143 17)(3 144 18)(4 133 19)(5 134 20)(6 135 21)(7 136 22)(8 137 23)(9 138 24)(10 139 13)(11 140 14)(12 141 15)(25 74 117)(26 75 118)(27 76 119)(28 77 120)(29 78 109)(30 79 110)(31 80 111)(32 81 112)(33 82 113)(34 83 114)(35 84 115)(36 73 116)(37 92 53)(38 93 54)(39 94 55)(40 95 56)(41 96 57)(42 85 58)(43 86 59)(44 87 60)(45 88 49)(46 89 50)(47 90 51)(48 91 52)(61 124 105)(62 125 106)(63 126 107)(64 127 108)(65 128 97)(66 129 98)(67 130 99)(68 131 100)(69 132 101)(70 121 102)(71 122 103)(72 123 104)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 97)(12 98)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 118)(38 119)(39 120)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 85)(81 86)(82 87)(83 88)(84 89)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 141)(130 142)(131 143)(132 144)
(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 90 7 96)(2 89 8 95)(3 88 9 94)(4 87 10 93)(5 86 11 92)(6 85 12 91)(13 38 19 44)(14 37 20 43)(15 48 21 42)(16 47 22 41)(17 46 23 40)(18 45 24 39)(25 123 31 129)(26 122 32 128)(27 121 33 127)(28 132 34 126)(29 131 35 125)(30 130 36 124)(49 138 55 144)(50 137 56 143)(51 136 57 142)(52 135 58 141)(53 134 59 140)(54 133 60 139)(61 110 67 116)(62 109 68 115)(63 120 69 114)(64 119 70 113)(65 118 71 112)(66 117 72 111)(73 105 79 99)(74 104 80 98)(75 103 81 97)(76 102 82 108)(77 101 83 107)(78 100 84 106)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 4F 6A ··· 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C ··· 12N 12O ··· 12V 12W 12X order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 9 9 2 ··· 2 4 4 4 4 2 6 6 18 54 54 2 ··· 2 4 4 4 4 18 18 2 2 4 ··· 4 12 ··· 12 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 Q8 D6 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 S3×Dic6 kernel C3⋊S3×Dic6 Dic3×C3⋊S3 C33⋊4Q8 C32×Dic6 C12×C3⋊S3 C33⋊8Q8 C3×Dic6 C4×C3⋊S3 C3×C3⋊S3 C3×Dic3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C3 # reps 1 2 2 1 1 1 4 1 2 8 1 5 1 4 4 4 4 8

Matrix representation of C3⋊S3×Dic6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 3 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 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 1 10 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 9 4 0 0 0 0 0 0 6 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 9 9 0 0 0 0 0 0 1 4 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 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 6 6 0 0 0 0 0 0 9 7 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,6,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,1,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[6,9,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3⋊S3×Dic6 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times {\rm Dic}_6
% in TeX

G:=Group("C3:S3xDic6");
// GroupNames label

G:=SmallGroup(432,663);
// by ID

G=gap.SmallGroup(432,663);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,571,2028,14118]);
// Polycyclic

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

׿
×
𝔽