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G = C2×C6×Dic9order 432 = 24·33

Direct product of C2×C6 and Dic9

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

Aliases: C2×C6×Dic9, C62.132D6, C62.21Dic3, (C6×C18)⋊6C4, C185(C2×C12), (C2×C18)⋊11C12, C95(C22×C12), (C2×C6).54D18, C23.4(C3×D9), (C2×C62).23S3, C6.21(C6×Dic3), C6.57(C22×D9), C22.11(C6×D9), (C22×C6).12D9, C18.23(C22×C6), (C6×C18).46C22, (C3×C18).46C23, (C22×C18).15C6, C32.3(C22×Dic3), C2.2(C2×C6×D9), C6.43(S3×C2×C6), (C2×C6×C18).6C2, (C3×C18)⋊8(C2×C4), (C3×C9)⋊9(C22×C4), C3.1(Dic3×C2×C6), (C2×C6).61(S3×C6), (C2×C18).32(C2×C6), (C22×C6).27(C3×S3), (C2×C6).21(C3×Dic3), (C3×C6).64(C2×Dic3), (C3×C6).160(C22×S3), SmallGroup(432,372)

Series: Derived Chief Lower central Upper central

C1C9 — C2×C6×Dic9
C1C3C9C18C3×C18C3×Dic9C6×Dic9 — C2×C6×Dic9
C9 — C2×C6×Dic9
C1C22×C6

Generators and relations for C2×C6×Dic9
 G = < a,b,c,d | a2=b6=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 446 in 194 conjugacy classes, 118 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×C18, C2×C6×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, D9, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, Dic9, D18, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C3×D9, C2×Dic9, C22×D9, C6×Dic3, S3×C2×C6, C3×Dic9, C6×D9, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×D9, C2×C6×Dic9

Smallest permutation representation of C2×C6×Dic9
On 144 points
Generators in S144
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 125)(74 126)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 127)(108 128)
(1 34 13 28 7 22)(2 35 14 29 8 23)(3 36 15 30 9 24)(4 19 16 31 10 25)(5 20 17 32 11 26)(6 21 18 33 12 27)(37 70 49 64 43 58)(38 71 50 65 44 59)(39 72 51 66 45 60)(40 55 52 67 46 61)(41 56 53 68 47 62)(42 57 54 69 48 63)(73 103 79 91 85 97)(74 104 80 92 86 98)(75 105 81 93 87 99)(76 106 82 94 88 100)(77 107 83 95 89 101)(78 108 84 96 90 102)(109 143 115 131 121 137)(110 144 116 132 122 138)(111 127 117 133 123 139)(112 128 118 134 124 140)(113 129 119 135 125 141)(114 130 120 136 126 142)
(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 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 97 28 106)(20 96 29 105)(21 95 30 104)(22 94 31 103)(23 93 32 102)(24 92 33 101)(25 91 34 100)(26 108 35 99)(27 107 36 98)(37 113 46 122)(38 112 47 121)(39 111 48 120)(40 110 49 119)(41 109 50 118)(42 126 51 117)(43 125 52 116)(44 124 53 115)(45 123 54 114)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)

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

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

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

144 conjugacy classes

class 1 2A···2G3A3B3C3D3E4A···4H6A···6N6O···6AI9A···9I12A···12P18A···18BK
order12···2333334···46···66···69···912···1218···18
size11···1112229···91···12···22···29···92···2

144 irreducible representations

dim11111111222222222222
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6D9C3×S3Dic9D18C3×Dic3S3×C6C3×D9C3×Dic9C6×D9
kernelC2×C6×Dic9C6×Dic9C2×C6×C18C22×Dic9C6×C18C2×Dic9C22×C18C2×C18C2×C62C62C62C22×C6C22×C6C2×C6C2×C6C2×C6C2×C6C23C22C22
# reps1612812216143321298662418

Matrix representation of C2×C6×Dic9 in GL4(𝔽37) generated by

1000
03600
00360
00036
,
11000
02700
00100
00010
,
36000
0100
00160
00147
,
31000
0100
00239
00314
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[11,0,0,0,0,27,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,16,14,0,0,0,7],[31,0,0,0,0,1,0,0,0,0,23,3,0,0,9,14] >;

C2×C6×Dic9 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_9
% in TeX

G:=Group("C2xC6xDic9");
// GroupNames label

G:=SmallGroup(432,372);
// by ID

G=gap.SmallGroup(432,372);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^18=1,d^2=c^9,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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