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G = C22×C32⋊C12  order 432 = 24·33

Direct product of C22 and C32⋊C12

direct product, metabelian, supersoluble, monomial

Aliases: C22×C32⋊C12, C62⋊5C12, C62⋊7Dic3, C62.41D6, (C2×C62).7C6, He3⋊7(C22×C4), C62.13(C2×C6), (C2×C62).10S3, (C22×He3)⋊6C4, C6.22(C6×Dic3), C32⋊2(C22×C12), (C23×He3).4C2, C23.4(C32⋊C6), (C2×He3).27C23, C32⋊2(C22×Dic3), (C22×He3).30C22, C6.45(S3×C2×C6), (C3×C6)⋊2(C2×C12), C3.2(Dic3×C2×C6), (C2×C6).63(S3×C6), (C2×He3)⋊6(C2×C4), C3⋊Dic3⋊5(C2×C6), (C2×C3⋊Dic3)⋊6C6, (C3×C6)⋊2(C2×Dic3), (C3×C6).9(C22×C6), (C22×C6).29(C3×S3), (C3×C6).35(C22×S3), (C22×C3⋊Dic3)⋊2C3, (C2×C6).22(C3×Dic3), C2.2(C22×C32⋊C6), C22.11(C2×C32⋊C6), SmallGroup(432,376)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C22×C32⋊C12
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C32⋊C12 — C2×C32⋊C12 — C22×C32⋊C12
Lower central
C32 — C22×C32⋊C12
Upper central
C1 — C23

Generators and relations for C22×C32⋊C12
 G = < a,b,c,d,e | a2=b2=c3=d3=e12=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d, ede-1=d-1 >

Subgroups: 689 in 221 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, He3, C3×Dic3, C3⋊Dic3, C62, C62, C22×Dic3, C22×C12, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C32⋊C12, C22×He3, Dic3×C2×C6, C22×C3⋊Dic3, C2×C32⋊C12, C23×He3, C22×C32⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C32⋊C6, C6×Dic3, S3×C2×C6, C32⋊C12, C2×C32⋊C6, Dic3×C2×C6, C2×C32⋊C12, C22×C32⋊C6, C22×C32⋊C12

Smallest permutation representation of C22×C32⋊C12
►On 144 points
Generators in S144
(1 16)(2 13)(3 14)(4 15)(5 46)(6 47)(7 48)(8 45)(9 32)(10 29)(11 30)(12 31)(17 42)(18 43)(19 44)(20 41)(21 27)(22 28)(23 25)(24 26)(33 40)(34 37)(35 38)(36 39)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 85)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 97)(69 98)(70 99)(71 100)(72 101)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 141)(81 142)(82 143)(83 144)(84 133)(109 128)(110 129)(111 130)(112 131)(113 132)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)

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

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

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

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

80 conjugacy classes

class 1 2A···2G3A3B3C3D3E3F4A···4H6A···6G6H···6U6V···6AP12A···12P
order12···23333334···46···66···66···612···12
size11···12336669···92···23···36···69···9

80 irreducible representations

dim11111111222222666
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6C32⋊C6C32⋊C12C2×C32⋊C6
kernelC22×C32⋊C12C2×C32⋊C12C23×He3C22×C3⋊Dic3C22×He3C2×C3⋊Dic3C2×C62C62C2×C62C62C62C22×C6C2×C6C2×C6C23C22C22
# reps1612812216143286143

Matrix representation of C22×C32⋊C12 ►in GL10(𝔽13)

1000000000
0100000000
00120000000
00012000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
12000000000
01200000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
01200000000
11200000000
001212000000
0010000000
00000000012
00000000112
00000120000
00001120000
00000001200
00000011200
,
1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00000012100
00000012000
00000000121
00000000120
,
8500000000
0500000000
0070000000
0066000000
0000850000
0000050000
0000000050
0000000058
0000000800
0000008000

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,7,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,8,0,0] >;
 

C22×C32⋊C12 in GAP, Magma, Sage, TeX 

C_2^2\times C_3^2\rtimes C_{12}
 
% in TeX
 
G:=Group("C2^2xC3^2:C12");
 
// GroupNames label
 
G:=SmallGroup(432,376);
 
// by ID
 
G=gap.SmallGroup(432,376);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,4037,1034,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^12=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,e*c*e^-1=c^-1*d,e*d*e^-1=d^-1>;
 
// generators/relations
 

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