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## G = S3×C2×C36order 432 = 24·33

### Direct product of C2×C36 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C36
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C2×C18 — S3×C2×C36
 Lower central C3 — S3×C2×C36
 Upper central C1 — C2×C36

Generators and relations for S3×C2×C36
G = < a,b,c,d | a2=b36=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 308 in 178 conjugacy classes, 105 normal (33 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, S3×C9, C3×C18, C3×C18, C2×C36, C2×C36, C22×C18, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C9×Dic3, C3×C36, S3×C18, C6×C18, C22×C36, S3×C2×C12, S3×C36, Dic3×C18, C6×C36, S3×C2×C18, S3×C2×C36
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C9, C12, D6, C2×C6, C22×C4, C18, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C36, C2×C18, S3×C6, S3×C2×C4, C22×C12, S3×C9, C2×C36, C22×C18, S3×C12, S3×C2×C6, S3×C18, C22×C36, S3×C2×C12, S3×C36, S3×C2×C18, S3×C2×C36

Smallest permutation representation of S3×C2×C36
On 144 points
Generators in S144
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(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 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 108)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)

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

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

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

216 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P ··· 6W 9A ··· 9F 9G ··· 9L 12A ··· 12H 12I ··· 12T 12U ··· 12AB 18A ··· 18R 18S ··· 18AJ 18AK ··· 18BH 36A ··· 36X 36Y ··· 36AV 36AW ··· 36BT order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C9 C12 C18 C18 C18 C18 C36 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C9 S3×C12 S3×C18 S3×C18 S3×C36 kernel S3×C2×C36 S3×C36 Dic3×C18 C6×C36 S3×C2×C18 S3×C2×C12 S3×C18 S3×C12 C6×Dic3 C6×C12 S3×C2×C6 S3×C2×C4 S3×C6 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C36 C36 C2×C18 C2×C12 C18 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 6 16 24 6 6 6 48 1 2 1 2 4 4 2 6 8 12 6 24

Matrix representation of S3×C2×C36 in GL3(𝔽37) generated by

 36 0 0 0 1 0 0 0 1
,
 33 0 0 0 19 0 0 0 19
,
 1 0 0 0 10 10 0 0 26
,
 36 0 0 0 1 0 0 9 36
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[33,0,0,0,19,0,0,0,19],[1,0,0,0,10,0,0,10,26],[36,0,0,0,1,9,0,0,36] >;

S3×C2×C36 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{36}
% in TeX

G:=Group("S3xC2xC36");
// GroupNames label

G:=SmallGroup(432,345);
// by ID

G=gap.SmallGroup(432,345);
# by ID

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

G:=Group<a,b,c,d|a^2=b^36=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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