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## G = C2×S3×Dic9order 432 = 24·33

### Direct product of C2, S3 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×S3×Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — S3×Dic9 — C2×S3×Dic9
 Lower central C3×C9 — C2×S3×Dic9
 Upper central C1 — C22

Generators and relations for C2×S3×Dic9
G = < a,b,c,d,e | a2=b3=c2=d18=1, e2=d9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 748 in 178 conjugacy classes, 77 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, Dic9, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S3×C6, C62, S3×C2×C4, C22×Dic3, S3×C9, C3×C18, C3×C18, C2×Dic9, C2×Dic9, C22×C18, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C3×Dic9, C9⋊Dic3, S3×C18, C6×C18, C22×Dic9, C2×S3×Dic3, S3×Dic9, C6×Dic9, C2×C9⋊Dic3, S3×C2×C18, C2×S3×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C4×S3, C2×Dic3, C22×S3, Dic9, D18, S32, S3×C2×C4, C22×Dic3, C2×Dic9, C22×D9, S3×Dic3, C2×S32, S3×D9, C22×Dic9, C2×S3×Dic3, S3×Dic9, C2×S3×D9, C2×S3×Dic9

Smallest permutation representation of C2×S3×Dic9
On 144 points
Generators in S144
(1 32)(2 33)(3 34)(4 35)(5 36)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)
(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 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(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)(109 115 121)(110 116 122)(111 117 123)(112 118 124)(113 119 125)(114 120 126)(127 133 139)(128 134 140)(129 135 141)(130 136 142)(131 137 143)(132 138 144)
(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 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 91)(33 92)(34 93)(35 94)(36 95)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 109)(55 142)(56 143)(57 144)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)
(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 47 10 38)(2 46 11 37)(3 45 12 54)(4 44 13 53)(5 43 14 52)(6 42 15 51)(7 41 16 50)(8 40 17 49)(9 39 18 48)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 63 35 72)(27 62 36 71)(73 126 82 117)(74 125 83 116)(75 124 84 115)(76 123 85 114)(77 122 86 113)(78 121 87 112)(79 120 88 111)(80 119 89 110)(81 118 90 109)(91 144 100 135)(92 143 101 134)(93 142 102 133)(94 141 103 132)(95 140 104 131)(96 139 105 130)(97 138 106 129)(98 137 107 128)(99 136 108 127)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A ··· 18I 18J ··· 18R 18S ··· 18AD order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 3 3 2 2 4 9 9 9 9 27 27 27 27 2 ··· 2 4 4 4 6 6 6 6 2 2 2 4 4 4 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + + + - + + + - + + - + image C1 C2 C2 C2 C2 C4 S3 S3 D6 D6 Dic3 D6 D6 D9 C4×S3 Dic9 D18 D18 S32 S3×Dic3 C2×S32 S3×D9 S3×Dic9 C2×S3×D9 kernel C2×S3×Dic9 S3×Dic9 C6×Dic9 C2×C9⋊Dic3 S3×C2×C18 S3×C18 C2×Dic9 S3×C2×C6 Dic9 C2×C18 S3×C6 S3×C6 C62 C22×S3 C18 D6 D6 C2×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 1 2 1 4 2 1 3 4 12 6 3 1 2 1 3 6 3

Matrix representation of C2×S3×Dic9 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 36 1 0 0 36 0 0 0 0 0 1 0 0 0 0 1
,
 0 36 0 0 36 0 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 36 0 0 0 0 11 20 0 0 17 31
,
 6 0 0 0 0 6 0 0 0 0 36 0 0 0 36 1
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[36,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,11,17,0,0,20,31],[6,0,0,0,0,6,0,0,0,0,36,36,0,0,0,1] >;

C2×S3×Dic9 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_9
% in TeX

G:=Group("C2xS3xDic9");
// GroupNames label

G:=SmallGroup(432,308);
// by ID

G=gap.SmallGroup(432,308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^18=1,e^2=d^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽