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## G = S3×C22×C18order 432 = 24·33

### Direct product of C22×C18 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C22×C18
 Chief series C1 — C3 — C32 — C3×C9 — S3×C9 — S3×C18 — S3×C2×C18 — S3×C22×C18
 Lower central C3 — S3×C22×C18
 Upper central C1 — C22×C18

Generators and relations for S3×C22×C18
G = < a,b,c,d,e | a2=b2=c18=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 772 in 434 conjugacy classes, 249 normal (15 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C9, C9, C32, D6, C2×C6, C2×C6, C24, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, C22×C6, C3×C9, C2×C18, C2×C18, S3×C6, C62, S3×C23, C23×C6, S3×C9, C3×C18, C22×C18, C22×C18, S3×C2×C6, C2×C62, S3×C18, C6×C18, C23×C18, S3×C22×C6, S3×C2×C18, C2×C6×C18, S3×C22×C18
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2×C6, C24, C18, C3×S3, C22×S3, C22×C6, C2×C18, S3×C6, S3×C23, C23×C6, S3×C9, C22×C18, S3×C2×C6, S3×C18, C23×C18, S3×C22×C6, S3×C2×C18, S3×C22×C18

Smallest permutation representation of S3×C22×C18
On 144 points
Generators in S144
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 55)(16 56)(17 57)(18 58)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 126)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 19)(37 69)(38 70)(39 71)(40 72)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(73 105)(74 106)(75 107)(76 108)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(109 141)(110 142)(111 143)(112 144)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)
(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 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)(109 121 115)(110 122 116)(111 123 117)(112 124 118)(113 125 119)(114 126 120)(127 139 133)(128 140 134)(129 141 135)(130 142 136)(131 143 137)(132 144 138)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 104)(20 105)(21 106)(22 107)(23 108)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 109)(51 110)(52 111)(53 112)(54 113)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 127)(69 128)(70 129)(71 130)(72 131)

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

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

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

216 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY 9A ··· 9F 9G ··· 9L 18A ··· 18AP 18AQ ··· 18CF 18CG ··· 18EB order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 ··· 1 3 ··· 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 C3×S3 S3×C6 S3×C9 S3×C18 kernel S3×C22×C18 S3×C2×C18 C2×C6×C18 S3×C22×C6 S3×C2×C6 C2×C62 S3×C23 C22×S3 C22×C6 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 # reps 1 14 1 2 28 2 6 84 6 1 7 2 14 6 42

Matrix representation of S3×C22×C18 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1
,
 4 0 0 0 0 0 13 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 7
,
 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,13,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7],[18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

S3×C22×C18 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{18}
% in TeX

G:=Group("S3xC2^2xC18");
// GroupNames label

G:=SmallGroup(432,557);
// by ID

G=gap.SmallGroup(432,557);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,137,14118]);
// Polycyclic

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

׿
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