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G = C62⋊3C12  order 432 = 24·33

2nd semidirect product of C62 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: C62⋊3C12, C62⋊3Dic3, C62.26D6, C62⋊5C4⋊C3, (C2×C62).2S3, (C2×C62).4C6, C62.8(C2×C6), He3⋊7(C22⋊C4), (C2×He3).31D4, (C22×He3)⋊4C4, C6.19(C6×Dic3), C2.3(He3⋊6D4), (C23×He3).2C2, C22⋊3(C32⋊C12), C23.3(C32⋊C6), C32⋊3(C6.D4), (C22×He3).20C22, (C3×C6).9(C2×C12), (C2×C6).47(S3×C6), (C2×C3⋊Dic3)⋊2C6, (C3×C6).16(C3×D4), C6.31(C3×C3⋊D4), (C2×C32⋊C12)⋊4C2, C2.5(C2×C32⋊C12), C32⋊3(C3×C22⋊C4), (C3×C6).31(C3⋊D4), (C2×He3).30(C2×C4), (C22×C6).25(C3×S3), (C2×C6).19(C3×Dic3), (C3×C6).14(C2×Dic3), C3.2(C3×C6.D4), C22.7(C2×C32⋊C6), SmallGroup(432,166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62⋊3C12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C2×C32⋊C12 — C62⋊3C12
Lower central
C32 — C3×C6 — C62⋊3C12
Upper central
C1 — C22 — C23

Generators and relations for C62⋊3C12
 G = < a,b,c | a6=b6=c12=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >

Subgroups: 549 in 151 conjugacy classes, 46 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, C2×C6, 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, C62, C6.D4, C3×C22⋊C4, C2×He3, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C32⋊C12, C22×He3, C22×He3, C22×He3, C3×C6.D4, C62⋊5C4, C2×C32⋊C12, C23×He3, C62⋊3C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C32⋊C6, C6×Dic3, C3×C3⋊D4, C32⋊C12, C2×C32⋊C6, C3×C6.D4, C2×C32⋊C12, He3⋊6D4, C62⋊3C12

Smallest permutation representation of C62⋊3C12
►On 72 points
Generators in S72
(1 11 33 7 5 27)(2 53 30 57 10 45)(3 72 70 9 66 64)(4 39 8 59 36 51)(6 15 69 49 61 23)(12 21 63 55 67 17)(13 32 43 71 47 28)(14 24 52 20 18 58)(16 40 38 22 46 44)(19 26 37 65 41 34)(25 35 68 31 29 62)(42 56 54 48 50 60)

(1 20 25 50 72 46)(2 47 61 51 26 21)(3 22 27 52 62 48)(4 37 63 53 28 23)(5 24 29 54 64 38)(6 39 65 55 30 13)(7 14 31 56 66 40)(8 41 67 57 32 15)(9 16 33 58 68 42)(10 43 69 59 34 17)(11 18 35 60 70 44)(12 45 71 49 36 19)

(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)
 
G:=sub<Sym(72)| (1,11,33,7,5,27)(2,53,30,57,10,45)(3,72,70,9,66,64)(4,39,8,59,36,51)(6,15,69,49,61,23)(12,21,63,55,67,17)(13,32,43,71,47,28)(14,24,52,20,18,58)(16,40,38,22,46,44)(19,26,37,65,41,34)(25,35,68,31,29,62)(42,56,54,48,50,60), (1,20,25,50,72,46)(2,47,61,51,26,21)(3,22,27,52,62,48)(4,37,63,53,28,23)(5,24,29,54,64,38)(6,39,65,55,30,13)(7,14,31,56,66,40)(8,41,67,57,32,15)(9,16,33,58,68,42)(10,43,69,59,34,17)(11,18,35,60,70,44)(12,45,71,49,36,19), (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)>;
 
G:=Group( (1,11,33,7,5,27)(2,53,30,57,10,45)(3,72,70,9,66,64)(4,39,8,59,36,51)(6,15,69,49,61,23)(12,21,63,55,67,17)(13,32,43,71,47,28)(14,24,52,20,18,58)(16,40,38,22,46,44)(19,26,37,65,41,34)(25,35,68,31,29,62)(42,56,54,48,50,60), (1,20,25,50,72,46)(2,47,61,51,26,21)(3,22,27,52,62,48)(4,37,63,53,28,23)(5,24,29,54,64,38)(6,39,65,55,30,13)(7,14,31,56,66,40)(8,41,67,57,32,15)(9,16,33,58,68,42)(10,43,69,59,34,17)(11,18,35,60,70,44)(12,45,71,49,36,19), (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) );
 
G=PermutationGroup([[(1,11,33,7,5,27),(2,53,30,57,10,45),(3,72,70,9,66,64),(4,39,8,59,36,51),(6,15,69,49,61,23),(12,21,63,55,67,17),(13,32,43,71,47,28),(14,24,52,20,18,58),(16,40,38,22,46,44),(19,26,37,65,41,34),(25,35,68,31,29,62),(42,56,54,48,50,60)], [(1,20,25,50,72,46),(2,47,61,51,26,21),(3,22,27,52,62,48),(4,37,63,53,28,23),(5,24,29,54,64,38),(6,39,65,55,30,13),(7,14,31,56,66,40),(8,41,67,57,32,15),(9,16,33,58,68,42),(10,43,69,59,34,17),(11,18,35,60,70,44),(12,45,71,49,36,19)], [(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)]])
 

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A···6G6H···6M6N···6AL12A···12H
order12222233333344446···66···66···612···12
size111122233666181818182···23···36···618···18

62 irreducible representations

dim1111111122222222226666
type+++++-++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3×S3C3⋊D4C3×D4C3×Dic3S3×C6C3×C3⋊D4C32⋊C6C32⋊C12C2×C32⋊C6He3⋊6D4
kernelC62⋊3C12C2×C32⋊C12C23×He3C62⋊5C4C22×He3C2×C3⋊Dic3C2×C62C62C2×C62C2×He3C62C62C22×C6C3×C6C3×C6C2×C6C2×C6C6C23C22C22C2
# reps1212442812212444281214

Matrix representation of C62⋊3C12 ►in GL8(𝔽13)

90000000
03000000
000012000
001200000
000120000
00000010
00000001
00000100
,
10000000
01000000
001000000
000100000
000010000
00000400
00000040
00000004
,
01000000
120000000
00000009
00000100
00000030
00004000
001200000
000100000

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0] >;
 

C62⋊3C12 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_3C_{12}
 
% in TeX
 
G:=Group("C6^2:3C12");
 
// GroupNames label
 
G:=SmallGroup(432,166);
 
// by ID
 
G=gap.SmallGroup(432,166);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,2035,14118]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^6=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
 
// generators/relations
 

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