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G = S3×C72order 432 = 24·33

Direct product of C72 and S3

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

Aliases: S3×C72, C244C18, D6.2C36, C36.76D6, Dic3.2C36, C3⋊C86C18, C31(C2×C72), (C3×C72)⋊1C2, (S3×C24).C3, (C3×S3).C24, C2.1(S3×C36), C3.4(S3×C24), C6.1(C2×C36), (S3×C6).6C12, (S3×C36).6C2, (S3×C18).4C4, (C4×S3).3C18, C24.27(C3×S3), C6.33(S3×C12), C18.21(C4×S3), C4.12(S3×C18), (C3×C24).20C6, (S3×C12).12C6, C12.116(S3×C6), C12.12(C2×C18), C32.2(C2×C24), (C9×Dic3).4C4, (C3×C36).50C22, (C3×Dic3).6C12, (C3×C9)⋊4(C2×C8), (C9×C3⋊C8)⋊13C2, (C3×C3⋊C8).9C6, (C3×C6).37(C2×C12), (C3×C18).14(C2×C4), (C3×C12).87(C2×C6), SmallGroup(432,109)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C72
C1C3C6C3×C6C3×C12C3×C36S3×C36 — S3×C72
C3 — S3×C72
C1C72

Generators and relations for S3×C72
 G = < a,b,c | a72=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 74 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4, C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C2×C8, C18, C18 [×3], C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C3×C9, C36, C36 [×2], C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, S3×C9 [×2], C3×C18, C72, C72 [×2], C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C2×C72, S3×C24, C9×C3⋊C8, C3×C72, S3×C36, S3×C72
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C9, C12 [×2], D6, C2×C6, C2×C8, C18 [×3], C3×S3, C24 [×2], C4×S3, C2×C12, C36 [×2], C2×C18, S3×C6, S3×C8, C2×C24, S3×C9, C72 [×2], C2×C36, S3×C12, S3×C18, C2×C72, S3×C24, S3×C36, S3×C72

Smallest permutation representation of S3×C72
On 144 points
Generators in S144
(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 25 49)(2 26 50)(3 27 51)(4 28 52)(5 29 53)(6 30 54)(7 31 55)(8 32 56)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 41 65)(18 42 66)(19 43 67)(20 44 68)(21 45 69)(22 46 70)(23 47 71)(24 48 72)(73 121 97)(74 122 98)(75 123 99)(76 124 100)(77 125 101)(78 126 102)(79 127 103)(80 128 104)(81 129 105)(82 130 106)(83 131 107)(84 132 108)(85 133 109)(86 134 110)(87 135 111)(88 136 112)(89 137 113)(90 138 114)(91 139 115)(92 140 116)(93 141 117)(94 142 118)(95 143 119)(96 144 120)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)

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

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

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

216 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I8A8B8C8D8E8F8G8H9A···9F9G···9L12A12B12C12D12E···12J12K12L12M12N18A···18F18G···18L18M···18X24A···24H24I···24T24U···24AB36A···36L36M···36X36Y···36AJ72A···72X72Y···72AV72AW···72BT
order1222333334444666666666888888889···99···91212121212···121212121218···1818···1818···1824···2424···2424···2436···3636···3636···3672···7272···7272···72
size1133112221133112223333111133331···12···211112···233331···12···23···31···12···23···31···12···23···31···12···23···3

216 irreducible representations

dim111111111111111111111222222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C9C12C12C18C18C18C24C36C36C72S3D6C3×S3C4×S3S3×C6S3×C8S3×C9S3×C12S3×C18S3×C24S3×C36S3×C72
kernelS3×C72C9×C3⋊C8C3×C72S3×C36S3×C24C9×Dic3S3×C18C3×C3⋊C8C3×C24S3×C12S3×C9S3×C8C3×Dic3S3×C6C3⋊C8C24C4×S3C3×S3Dic3D6S3C72C36C24C18C12C9C8C6C4C3C2C1
# reps111122222286446661612124811222464681224

Matrix representation of S3×C72 in GL2(𝔽73) generated by

50
05
,
88
064
,
10
772
G:=sub<GL(2,GF(73))| [5,0,0,5],[8,0,8,64],[1,7,0,72] >;

S3×C72 in GAP, Magma, Sage, TeX

S_3\times C_{72}
% in TeX

G:=Group("S3xC72");
// GroupNames label

G:=SmallGroup(432,109);
// by ID

G=gap.SmallGroup(432,109);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,92,142,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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