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G = C3×C12⋊Dic3order 432 = 24·33

Direct product of C3 and C12⋊Dic3

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12⋊Dic3, C62.142D6, (C3×C12)⋊7C12, C3319(C4⋊C4), (C6×C12).28C6, (C6×C12).44S3, (C32×C12)⋊7C4, (C3×C12)⋊8Dic3, C121(C3×Dic3), (C3×C6).64D12, C6.22(C3×D12), C123(C3⋊Dic3), C62.66(C2×C6), (C3×C6).29Dic6, C6.27(C6×Dic3), C6.13(C3×Dic6), (C32×C6).60D4, (C32×C6).13Q8, C6.25(C12⋊S3), C3211(C4⋊Dic3), (C3×C62).48C22, C6.13(C324Q8), C4⋊(C3×C3⋊Dic3), (C3×C6×C12).10C2, C32(C3×C4⋊Dic3), C3212(C3×C4⋊C4), (C2×C6).69(S3×C6), (C3×C6).52(C3×D4), C22.5(C6×C3⋊S3), C2.4(C6×C3⋊Dic3), (C3×C6).15(C3×Q8), C2.1(C3×C12⋊S3), (C2×C12).20(C3×S3), (C3×C6).64(C2×C12), (C2×C3⋊Dic3).9C6, (C6×C3⋊Dic3).5C2, C6.23(C2×C3⋊Dic3), (C2×C12).26(C3⋊S3), C2.2(C3×C324Q8), (C32×C6).70(C2×C4), (C3×C6).69(C2×Dic3), (C2×C4).3(C3×C3⋊S3), (C2×C6).63(C2×C3⋊S3), SmallGroup(432,489)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×C12⋊Dic3
C1C3C32C3×C6C62C3×C62C6×C3⋊Dic3 — C3×C12⋊Dic3
C32C3×C6 — C3×C12⋊Dic3
C1C2×C6C2×C12

Generators and relations for C3×C12⋊Dic3
 G = < a,b,c,d | a3=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 532 in 220 conjugacy classes, 102 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C4⋊Dic3, C3×C4⋊C4, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, C3×C4⋊Dic3, C12⋊Dic3, C6×C3⋊Dic3, C3×C6×C12, C3×C12⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C4⋊C4, C3×C3⋊S3, C3×Dic6, C3×D12, C6×Dic3, C324Q8, C12⋊S3, C2×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C4⋊Dic3, C12⋊Dic3, C3×C324Q8, C3×C12⋊S3, C6×C3⋊Dic3, C3×C12⋊Dic3

Smallest permutation representation of C3×C12⋊Dic3
On 144 points
Generators in S144
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)(97 101 105)(98 102 106)(99 103 107)(100 104 108)(109 117 113)(110 118 114)(111 119 115)(112 120 116)(121 125 129)(122 126 130)(123 127 131)(124 128 132)(133 137 141)(134 138 142)(135 139 143)(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)
(1 70 41 18 109 95)(2 71 42 19 110 96)(3 72 43 20 111 85)(4 61 44 21 112 86)(5 62 45 22 113 87)(6 63 46 23 114 88)(7 64 47 24 115 89)(8 65 48 13 116 90)(9 66 37 14 117 91)(10 67 38 15 118 92)(11 68 39 16 119 93)(12 69 40 17 120 94)(25 129 54 136 77 98)(26 130 55 137 78 99)(27 131 56 138 79 100)(28 132 57 139 80 101)(29 121 58 140 81 102)(30 122 59 141 82 103)(31 123 60 142 83 104)(32 124 49 143 84 105)(33 125 50 144 73 106)(34 126 51 133 74 107)(35 127 52 134 75 108)(36 128 53 135 76 97)
(1 80 18 132)(2 79 19 131)(3 78 20 130)(4 77 21 129)(5 76 22 128)(6 75 23 127)(7 74 24 126)(8 73 13 125)(9 84 14 124)(10 83 15 123)(11 82 16 122)(12 81 17 121)(25 61 136 112)(26 72 137 111)(27 71 138 110)(28 70 139 109)(29 69 140 120)(30 68 141 119)(31 67 142 118)(32 66 143 117)(33 65 144 116)(34 64 133 115)(35 63 134 114)(36 62 135 113)(37 49 91 105)(38 60 92 104)(39 59 93 103)(40 58 94 102)(41 57 95 101)(42 56 96 100)(43 55 85 99)(44 54 86 98)(45 53 87 97)(46 52 88 108)(47 51 89 107)(48 50 90 106)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A3B3C···3N4A4B4C4D4E4F6A···6F6G···6AP12A···12AZ12BA···12BH
order1222333···34444446···66···612···1212···12
size1111112···222181818181···12···22···218···18

126 irreducible representations

dim1111111122222222222222
type+++++--+-+
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3×S3Dic6D12C3×D4C3×Q8C3×Dic3S3×C6C3×Dic6C3×D12
kernelC3×C12⋊Dic3C6×C3⋊Dic3C3×C6×C12C12⋊Dic3C32×C12C2×C3⋊Dic3C6×C12C3×C12C6×C12C32×C6C32×C6C3×C12C62C2×C12C3×C6C3×C6C3×C6C3×C6C12C2×C6C6C6
# reps1212442841184888221681616

Matrix representation of C3×C12⋊Dic3 in GL4(𝔽13) generated by

9000
0900
0090
0009
,
3000
0900
00110
0006
,
12000
01200
00100
0004
,
0100
12000
0001
00120
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,9,0,0,0,0,11,0,0,0,0,6],[12,0,0,0,0,12,0,0,0,0,10,0,0,0,0,4],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;

C3×C12⋊Dic3 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3xC12:Dic3");
// GroupNames label

G:=SmallGroup(432,489);
// by ID

G=gap.SmallGroup(432,489);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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