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## G = C2×C6×C3⋊Dic3order 432 = 24·33

### Direct product of C2×C6 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C6×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C6×C3⋊Dic3 — C2×C6×C3⋊Dic3
 Lower central C32 — C2×C6×C3⋊Dic3
 Upper central C1 — C22×C6

Generators and relations for C2×C6×C3⋊Dic3
G = < a,b,c,d,e | a2=b6=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 932 in 452 conjugacy classes, 214 normal (14 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C3 [×4], C4 [×4], C22 [×7], C6, C6 [×34], C6 [×28], C2×C4 [×6], C23, C32, C32 [×4], C32 [×4], Dic3 [×16], C12 [×4], C2×C6 [×35], C2×C6 [×28], C22×C4, C3×C6, C3×C6 [×34], C3×C6 [×28], C2×Dic3 [×24], C2×C12 [×6], C22×C6, C22×C6 [×4], C22×C6 [×4], C33, C3×Dic3 [×16], C3⋊Dic3 [×4], C62 [×35], C62 [×28], C22×Dic3 [×4], C22×C12, C32×C6, C32×C6 [×6], C6×Dic3 [×24], C2×C3⋊Dic3 [×6], C2×C62, C2×C62 [×4], C2×C62 [×4], C3×C3⋊Dic3 [×4], C3×C62 [×7], Dic3×C2×C6 [×4], C22×C3⋊Dic3, C6×C3⋊Dic3 [×6], C63, C2×C6×C3⋊Dic3
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3 [×4], C6 [×7], C2×C4 [×6], C23, Dic3 [×16], C12 [×4], D6 [×12], C2×C6 [×7], C22×C4, C3×S3 [×4], C3⋊S3, C2×Dic3 [×24], C2×C12 [×6], C22×S3 [×4], C22×C6, C3×Dic3 [×16], C3⋊Dic3 [×4], S3×C6 [×12], C2×C3⋊S3 [×3], C22×Dic3 [×4], C22×C12, C3×C3⋊S3, C6×Dic3 [×24], C2×C3⋊Dic3 [×6], S3×C2×C6 [×4], C22×C3⋊S3, C3×C3⋊Dic3 [×4], C6×C3⋊S3 [×3], Dic3×C2×C6 [×4], C22×C3⋊Dic3, C6×C3⋊Dic3 [×6], C2×C6×C3⋊S3, C2×C6×C3⋊Dic3

Smallest permutation representation of C2×C6×C3⋊Dic3
On 144 points
Generators in S144
(1 66)(2 61)(3 62)(4 63)(5 64)(6 65)(7 128)(8 129)(9 130)(10 131)(11 132)(12 127)(13 39)(14 40)(15 41)(16 42)(17 37)(18 38)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 125)(26 126)(27 121)(28 122)(29 123)(30 124)(31 116)(32 117)(33 118)(34 119)(35 120)(36 115)(43 82)(44 83)(45 84)(46 79)(47 80)(48 81)(49 88)(50 89)(51 90)(52 85)(53 86)(54 87)(55 94)(56 95)(57 96)(58 91)(59 92)(60 93)(73 112)(74 113)(75 114)(76 109)(77 110)(78 111)(97 136)(98 137)(99 138)(100 133)(101 134)(102 135)(103 142)(104 143)(105 144)(106 139)(107 140)(108 141)
(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 57 44)(2 58 45)(3 59 46)(4 60 47)(5 55 48)(6 56 43)(7 137 36)(8 138 31)(9 133 32)(10 134 33)(11 135 34)(12 136 35)(13 22 50)(14 23 51)(15 24 52)(16 19 53)(17 20 54)(18 21 49)(25 142 114)(26 143 109)(27 144 110)(28 139 111)(29 140 112)(30 141 113)(37 71 87)(38 72 88)(39 67 89)(40 68 90)(41 69 85)(42 70 86)(61 91 84)(62 92 79)(63 93 80)(64 94 81)(65 95 82)(66 96 83)(73 123 107)(74 124 108)(75 125 103)(76 126 104)(77 121 105)(78 122 106)(97 120 127)(98 115 128)(99 116 129)(100 117 130)(101 118 131)(102 119 132)
(1 15 46 54 55 22)(2 16 47 49 56 23)(3 17 48 50 57 24)(4 18 43 51 58 19)(5 13 44 52 59 20)(6 14 45 53 60 21)(7 27 34 114 133 140)(8 28 35 109 134 141)(9 29 36 110 135 142)(10 30 31 111 136 143)(11 25 32 112 137 144)(12 26 33 113 138 139)(37 81 89 96 69 62)(38 82 90 91 70 63)(39 83 85 92 71 64)(40 84 86 93 72 65)(41 79 87 94 67 66)(42 80 88 95 68 61)(73 98 105 132 125 117)(74 99 106 127 126 118)(75 100 107 128 121 119)(76 101 108 129 122 120)(77 102 103 130 123 115)(78 97 104 131 124 116)
(1 126 54 99)(2 121 49 100)(3 122 50 101)(4 123 51 102)(5 124 52 97)(6 125 53 98)(7 95 114 42)(8 96 109 37)(9 91 110 38)(10 92 111 39)(11 93 112 40)(12 94 113 41)(13 131 59 78)(14 132 60 73)(15 127 55 74)(16 128 56 75)(17 129 57 76)(18 130 58 77)(19 115 43 103)(20 116 44 104)(21 117 45 105)(22 118 46 106)(23 119 47 107)(24 120 48 108)(25 86 137 65)(26 87 138 66)(27 88 133 61)(28 89 134 62)(29 90 135 63)(30 85 136 64)(31 83 143 71)(32 84 144 72)(33 79 139 67)(34 80 140 68)(35 81 141 69)(36 82 142 70)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C ··· 3N 4A ··· 4H 6A ··· 6N 6O ··· 6CT 12A ··· 12P order 1 2 ··· 2 3 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel C2×C6×C3⋊Dic3 C6×C3⋊Dic3 C63 C22×C3⋊Dic3 C3×C62 C2×C3⋊Dic3 C2×C62 C62 C2×C62 C62 C62 C22×C6 C2×C6 C2×C6 # reps 1 6 1 2 8 12 2 16 4 16 12 8 32 24

Matrix representation of C2×C6×C3⋊Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 4 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 9 0 0 0 0 4 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 9 0 0 0 0 7 3
,
 1 0 0 0 0 0 12 8 0 0 0 3 1 0 0 0 0 0 12 12 0 0 0 0 1

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

C2×C6×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xC6xC3:Dic3");
// GroupNames label

G:=SmallGroup(432,718);
// by ID

G=gap.SmallGroup(432,718);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,4037,14118]);
// Polycyclic

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

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