Copied to
clipboard

## G = S3×C72order 432 = 24·33

### Direct product of C72 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C72
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — S3×C36 — S3×C72
 Lower central C3 — S3×C72
 Upper central C1 — C72

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 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 18A ··· 18F 18G ··· 18L 18M ··· 18X 24A ··· 24H 24I ··· 24T 24U ··· 24AB 36A ··· 36L 36M ··· 36X 36Y ··· 36AJ 72A ··· 72X 72Y ··· 72AV 72AW ··· 72BT order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 9 ··· 9 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 ··· 24 24 ··· 24 24 ··· 24 36 ··· 36 36 ··· 36 36 ··· 36 72 ··· 72 72 ··· 72 72 ··· 72 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C9 C12 C12 C18 C18 C18 C24 C36 C36 C72 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C9 S3×C12 S3×C18 S3×C24 S3×C36 S3×C72 kernel S3×C72 C9×C3⋊C8 C3×C72 S3×C36 S3×C24 C9×Dic3 S3×C18 C3×C3⋊C8 C3×C24 S3×C12 S3×C9 S3×C8 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 C3×S3 Dic3 D6 S3 C72 C36 C24 C18 C12 C9 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 6 4 4 6 6 6 16 12 12 48 1 1 2 2 2 4 6 4 6 8 12 24

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

 5 0 0 5
,
 8 8 0 64
,
 1 0 7 72
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

׿
×
𝔽