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G = Dic3×C2×C18order 432 = 24·33

Direct product of C2×C18 and Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: Dic3×C2×C18, C62.20C12, (C2×C6)⋊5C36, (C6×C18)⋊3C4, C62(C2×C36), C32(C22×C36), (C2×C18).54D6, C23.4(S3×C9), (C2×C62).24C6, (C22×C6).9C18, C62.57(C2×C6), C6.9(C22×C18), C6.34(C6×Dic3), (C22×C18).12S3, C22.11(S3×C18), C18.57(C22×S3), (C6×C18).30C22, (C3×C18).36C23, (C6×Dic3).17C6, C32.3(C22×C12), (C2×C6×C18).3C2, C2.2(S3×C2×C18), C6.70(S3×C2×C6), (C3×C18)⋊7(C2×C4), (C3×C9)⋊8(C22×C4), C3.4(Dic3×C2×C6), (C2×C6).88(S3×C6), (Dic3×C2×C6).2C3, (C2×C6).15(C2×C18), (C3×C6).55(C2×C12), (C3×C6).46(C22×C6), (C22×C6).38(C3×S3), (C2×C6).27(C3×Dic3), (C3×Dic3).21(C2×C6), SmallGroup(432,373)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C2×C18
C1C3C32C3×C6C3×C18C9×Dic3Dic3×C18 — Dic3×C2×C18
C3 — Dic3×C2×C18
C1C22×C18

Generators and relations for Dic3×C2×C18
 G = < a,b,c,d | a2=b18=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 292 in 194 conjugacy classes, 129 normal (21 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×4], C22 [×7], C6 [×2], C6 [×12], C6 [×7], C2×C4 [×6], C23, C9, C9, C32, Dic3 [×4], C12 [×4], C2×C6 [×14], C2×C6 [×7], C22×C4, C18, C18 [×6], C18 [×7], C3×C6, C3×C6 [×6], C2×Dic3 [×6], C2×C12 [×6], C22×C6 [×2], C22×C6, C3×C9, C36 [×4], C2×C18 [×7], C2×C18 [×7], C3×Dic3 [×4], C62 [×7], C22×Dic3, C22×C12, C3×C18, C3×C18 [×6], C2×C36 [×6], C22×C18, C22×C18, C6×Dic3 [×6], C2×C62, C9×Dic3 [×4], C6×C18 [×7], C22×C36, Dic3×C2×C6, Dic3×C18 [×6], C2×C6×C18, Dic3×C2×C18
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C9, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C18 [×7], C3×S3, C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C36 [×4], C2×C18 [×7], C3×Dic3 [×4], S3×C6 [×3], C22×Dic3, C22×C12, S3×C9, C2×C36 [×6], C22×C18, C6×Dic3 [×6], S3×C2×C6, C9×Dic3 [×4], S3×C18 [×3], C22×C36, Dic3×C2×C6, Dic3×C18 [×6], S3×C2×C18, Dic3×C2×C18

Smallest permutation representation of Dic3×C2×C18
On 144 points
Generators in S144
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 91)(87 92)(88 93)(89 94)(90 95)
(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 71 7 59 13 65)(2 72 8 60 14 66)(3 55 9 61 15 67)(4 56 10 62 16 68)(5 57 11 63 17 69)(6 58 12 64 18 70)(19 131 25 137 31 143)(20 132 26 138 32 144)(21 133 27 139 33 127)(22 134 28 140 34 128)(23 135 29 141 35 129)(24 136 30 142 36 130)(37 95 49 107 43 101)(38 96 50 108 44 102)(39 97 51 91 45 103)(40 98 52 92 46 104)(41 99 53 93 47 105)(42 100 54 94 48 106)(73 110 85 122 79 116)(74 111 86 123 80 117)(75 112 87 124 81 118)(76 113 88 125 82 119)(77 114 89 126 83 120)(78 115 90 109 84 121)
(1 92 59 40)(2 93 60 41)(3 94 61 42)(4 95 62 43)(5 96 63 44)(6 97 64 45)(7 98 65 46)(8 99 66 47)(9 100 67 48)(10 101 68 49)(11 102 69 50)(12 103 70 51)(13 104 71 52)(14 105 72 53)(15 106 55 54)(16 107 56 37)(17 108 57 38)(18 91 58 39)(19 75 137 124)(20 76 138 125)(21 77 139 126)(22 78 140 109)(23 79 141 110)(24 80 142 111)(25 81 143 112)(26 82 144 113)(27 83 127 114)(28 84 128 115)(29 85 129 116)(30 86 130 117)(31 87 131 118)(32 88 132 119)(33 89 133 120)(34 90 134 121)(35 73 135 122)(36 74 136 123)

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

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

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

216 conjugacy classes

class 1 2A···2G3A3B3C3D3E4A···4H6A···6N6O···6AI9A···9F9G···9L12A···12P18A···18AP18AQ···18CF36A···36AV
order12···2333334···46···66···69···99···912···1218···1818···1836···36
size11···1112223···31···12···21···12···23···31···12···23···3

216 irreducible representations

dim111111111111222222222
type++++-+
imageC1C2C2C3C4C6C6C9C12C18C18C36S3Dic3D6C3×S3C3×Dic3S3×C6S3×C9C9×Dic3S3×C18
kernelDic3×C2×C18Dic3×C18C2×C6×C18Dic3×C2×C6C6×C18C6×Dic3C2×C62C22×Dic3C62C2×Dic3C22×C6C2×C6C22×C18C2×C18C2×C18C22×C6C2×C6C2×C6C23C22C22
# reps161281226163664814328662418

Matrix representation of Dic3×C2×C18 in GL4(𝔽37) generated by

1000
03600
0010
0001
,
21000
0900
00120
00012
,
36000
03600
00110
00027
,
6000
0600
0001
00360
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,0,9,0,0,0,0,12,0,0,0,0,12],[36,0,0,0,0,36,0,0,0,0,11,0,0,0,0,27],[6,0,0,0,0,6,0,0,0,0,0,36,0,0,1,0] >;

Dic3×C2×C18 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{18}
% in TeX

G:=Group("Dic3xC2xC18");
// GroupNames label

G:=SmallGroup(432,373);
// by ID

G=gap.SmallGroup(432,373);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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