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G = C4×C9⋊C12order 432 = 24·33

Direct product of C4 and C9⋊C12

direct product, metacyclic, supersoluble, monomial

Aliases: C4×C9⋊C12, C362C12, Dic92C12, C62.22D6, 3- 1+2⋊C42, C9⋊(C4×C12), (C4×Dic9)⋊C3, (C2×C36).5C6, C6.17(S3×C12), C18.7(C2×C12), (C6×C12).19S3, C32.(C4×Dic3), (C2×Dic9).4C6, C3.3(Dic3×C12), C6.16(C6×Dic3), C12.16(C3×Dic3), (C3×C12).10Dic3, (C4×3- 1+2)⋊2C4, (C22×3- 1+2).2C22, C2.2(C4×C9⋊C6), C2.2(C2×C9⋊C12), (C2×C9⋊C12).4C2, (C2×C4).6(C9⋊C6), (C3×C6).17(C4×S3), (C2×C18).2(C2×C6), (C2×C6).42(S3×C6), C22.3(C2×C9⋊C6), (C2×C12).34(C3×S3), (C3×C6).12(C2×Dic3), (C2×C4×3- 1+2).5C2, (C2×3- 1+2).7(C2×C4), SmallGroup(432,144)

Series: Derived Chief Lower central Upper central

C1C9 — C4×C9⋊C12
C1C3C9C18C2×C18C22×3- 1+2C2×C9⋊C12 — C4×C9⋊C12
C9 — C4×C9⋊C12
C1C2×C4

Generators and relations for C4×C9⋊C12
 G = < a,b,c | a4=b9=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 262 in 98 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, 3- 1+2, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, C4×Dic9, Dic3×C12, C2×C9⋊C12, C2×C4×3- 1+2, C4×C9⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C9⋊C6, S3×C12, C6×Dic3, C9⋊C12, C2×C9⋊C6, Dic3×C12, C4×C9⋊C6, C2×C9⋊C12, C4×C9⋊C12

Smallest permutation representation of C4×C9⋊C12
On 144 points
Generators in S144
(1 11 26 46)(2 12 27 47)(3 9 28 48)(4 10 25 45)(5 13 43 31)(6 14 44 32)(7 15 41 29)(8 16 42 30)(17 38 34 23)(18 39 35 24)(19 40 36 21)(20 37 33 22)(49 131 65 91)(50 132 66 92)(51 121 67 93)(52 122 68 94)(53 123 69 95)(54 124 70 96)(55 125 71 85)(56 126 72 86)(57 127 61 87)(58 128 62 88)(59 129 63 89)(60 130 64 90)(73 120 141 107)(74 109 142 108)(75 110 143 97)(76 111 144 98)(77 112 133 99)(78 113 134 100)(79 114 135 101)(80 115 136 102)(81 116 137 103)(82 117 138 104)(83 118 139 105)(84 119 140 106)
(1 142 51 15 138 55 20 134 59)(2 56 143 17 52 135 16 60 139)(3 136 57 13 144 49 18 140 53)(4 50 137 19 58 141 14 54 133)(5 111 91 24 119 95 48 115 87)(6 96 112 45 92 116 21 88 120)(7 117 85 22 113 89 46 109 93)(8 90 118 47 86 110 23 94 114)(9 102 127 43 98 131 39 106 123)(10 132 103 40 128 107 44 124 99)(11 108 121 41 104 125 37 100 129)(12 126 97 38 122 101 42 130 105)(25 66 81 36 62 73 32 70 77)(26 74 67 29 82 71 33 78 63)(27 72 75 34 68 79 30 64 83)(28 80 61 31 76 65 35 84 69)
(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)| (1,11,26,46)(2,12,27,47)(3,9,28,48)(4,10,25,45)(5,13,43,31)(6,14,44,32)(7,15,41,29)(8,16,42,30)(17,38,34,23)(18,39,35,24)(19,40,36,21)(20,37,33,22)(49,131,65,91)(50,132,66,92)(51,121,67,93)(52,122,68,94)(53,123,69,95)(54,124,70,96)(55,125,71,85)(56,126,72,86)(57,127,61,87)(58,128,62,88)(59,129,63,89)(60,130,64,90)(73,120,141,107)(74,109,142,108)(75,110,143,97)(76,111,144,98)(77,112,133,99)(78,113,134,100)(79,114,135,101)(80,115,136,102)(81,116,137,103)(82,117,138,104)(83,118,139,105)(84,119,140,106), (1,142,51,15,138,55,20,134,59)(2,56,143,17,52,135,16,60,139)(3,136,57,13,144,49,18,140,53)(4,50,137,19,58,141,14,54,133)(5,111,91,24,119,95,48,115,87)(6,96,112,45,92,116,21,88,120)(7,117,85,22,113,89,46,109,93)(8,90,118,47,86,110,23,94,114)(9,102,127,43,98,131,39,106,123)(10,132,103,40,128,107,44,124,99)(11,108,121,41,104,125,37,100,129)(12,126,97,38,122,101,42,130,105)(25,66,81,36,62,73,32,70,77)(26,74,67,29,82,71,33,78,63)(27,72,75,34,68,79,30,64,83)(28,80,61,31,76,65,35,84,69), (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( (1,11,26,46)(2,12,27,47)(3,9,28,48)(4,10,25,45)(5,13,43,31)(6,14,44,32)(7,15,41,29)(8,16,42,30)(17,38,34,23)(18,39,35,24)(19,40,36,21)(20,37,33,22)(49,131,65,91)(50,132,66,92)(51,121,67,93)(52,122,68,94)(53,123,69,95)(54,124,70,96)(55,125,71,85)(56,126,72,86)(57,127,61,87)(58,128,62,88)(59,129,63,89)(60,130,64,90)(73,120,141,107)(74,109,142,108)(75,110,143,97)(76,111,144,98)(77,112,133,99)(78,113,134,100)(79,114,135,101)(80,115,136,102)(81,116,137,103)(82,117,138,104)(83,118,139,105)(84,119,140,106), (1,142,51,15,138,55,20,134,59)(2,56,143,17,52,135,16,60,139)(3,136,57,13,144,49,18,140,53)(4,50,137,19,58,141,14,54,133)(5,111,91,24,119,95,48,115,87)(6,96,112,45,92,116,21,88,120)(7,117,85,22,113,89,46,109,93)(8,90,118,47,86,110,23,94,114)(9,102,127,43,98,131,39,106,123)(10,132,103,40,128,107,44,124,99)(11,108,121,41,104,125,37,100,129)(12,126,97,38,122,101,42,130,105)(25,66,81,36,62,73,32,70,77)(26,74,67,29,82,71,33,78,63)(27,72,75,34,68,79,30,64,83)(28,80,61,31,76,65,35,84,69), (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([[(1,11,26,46),(2,12,27,47),(3,9,28,48),(4,10,25,45),(5,13,43,31),(6,14,44,32),(7,15,41,29),(8,16,42,30),(17,38,34,23),(18,39,35,24),(19,40,36,21),(20,37,33,22),(49,131,65,91),(50,132,66,92),(51,121,67,93),(52,122,68,94),(53,123,69,95),(54,124,70,96),(55,125,71,85),(56,126,72,86),(57,127,61,87),(58,128,62,88),(59,129,63,89),(60,130,64,90),(73,120,141,107),(74,109,142,108),(75,110,143,97),(76,111,144,98),(77,112,133,99),(78,113,134,100),(79,114,135,101),(80,115,136,102),(81,116,137,103),(82,117,138,104),(83,118,139,105),(84,119,140,106)], [(1,142,51,15,138,55,20,134,59),(2,56,143,17,52,135,16,60,139),(3,136,57,13,144,49,18,140,53),(4,50,137,19,58,141,14,54,133),(5,111,91,24,119,95,48,115,87),(6,96,112,45,92,116,21,88,120),(7,117,85,22,113,89,46,109,93),(8,90,118,47,86,110,23,94,114),(9,102,127,43,98,131,39,106,123),(10,132,103,40,128,107,44,124,99),(11,108,121,41,104,125,37,100,129),(12,126,97,38,122,101,42,130,105),(25,66,81,36,62,73,32,70,77),(26,74,67,29,82,71,33,78,63),(27,72,75,34,68,79,30,64,83),(28,80,61,31,76,65,35,84,69)], [(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 2A2B2C3A3B3C4A4B4C4D4E···4L6A6B6C6D···6I9A9B9C12A12B12C12D12E···12L12M···12AB18A···18I36A···36L
order122233344444···46666···69991212121212···1212···1218···1836···36
size111123311119···92223···366622223···39···96···66···6

80 irreducible representations

dim1111111111222222226666
type++++-++-+
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6C3×S3C4×S3C3×Dic3S3×C6S3×C12C9⋊C6C9⋊C12C2×C9⋊C6C4×C9⋊C6
kernelC4×C9⋊C12C2×C9⋊C12C2×C4×3- 1+2C4×Dic9C9⋊C12C4×3- 1+2C2×Dic9C2×C36Dic9C36C6×C12C3×C12C62C2×C12C3×C6C12C2×C6C6C2×C4C4C22C2
# reps12128442168121244281214

Matrix representation of C4×C9⋊C12 in GL10(𝔽37)

6000000000
0600000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
36100000000
36000000000
00361000000
00360000000
000011363500
00000013600
00000003601
0000110363636
00000003600
00001003600
,
12800000000
293600000000
002110000000
003116000000
0000840000
000012290000
0000124002533
000002900812
00008042900
0000124253300

G:=sub<GL(10,GF(37))| [6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,35,36,36,36,36,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0],[1,29,0,0,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,0,0,21,31,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,0,0,0,0,0,8,12,12,0,8,12,0,0,0,0,4,29,4,29,0,4,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,0,29,33,0,0,0,0,0,0,25,8,0,0,0,0,0,0,0,0,33,12,0,0] >;

C4×C9⋊C12 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes C_{12}
% in TeX

G:=Group("C4xC9:C12");
// GroupNames label

G:=SmallGroup(432,144);
// by ID

G=gap.SmallGroup(432,144);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^4=b^9=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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