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G = C624C12order 432 = 24·33

3rd semidirect product of C62 and C12 acting via C12/C2=C6

metabelian, soluble, monomial

Aliases: C624C12, C3⋊Dic3⋊A4, C32⋊(C4×A4), C6.9(S3×A4), (C6×A4).3S3, C32⋊A43C4, (C3×A4)⋊2Dic3, (C2×C62).6C6, C3.3(Dic3×A4), C222(C32⋊C12), C2.1(C62⋊C6), C23.2(C32⋊C6), (C3×C6).3(C2×A4), (C2×C32⋊A4).3C2, (C2×C6).7(C3×Dic3), (C22×C6).13(C3×S3), (C22×C3⋊Dic3)⋊1C3, SmallGroup(432,272)

Series: Derived Chief Lower central Upper central

C1C62 — C624C12
C1C3C32C62C2×C62C2×C32⋊A4 — C624C12
C62 — C624C12
C1C2

Generators and relations for C624C12
 G = < a,b,c | a6=b6=c12=1, ab=ba, cac-1=a2b, cbc-1=a3b-1 >

Subgroups: 533 in 92 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, C22 [×2], C6, C6 [×11], C2×C4 [×2], C23, C32, C32 [×2], Dic3 [×6], C12, A4 [×2], C2×C6, C2×C6 [×9], C22×C4, C3×C6, C3×C6 [×4], C2×Dic3 [×8], C2×A4 [×2], C22×C6, C22×C6, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×A4, C3×A4, C62, C62 [×2], C4×A4, C22×Dic3 [×2], C2×He3, C2×C3⋊Dic3 [×2], C6×A4, C6×A4, C2×C62, C32⋊C12, C32⋊A4, Dic3×A4, C22×C3⋊Dic3, C2×C32⋊A4, C624C12
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, C3×S3, C2×A4, C3×Dic3, C4×A4, C32⋊C6, S3×A4, C32⋊C12, Dic3×A4, C62⋊C6, C624C12

Smallest permutation representation of C624C12
On 36 points
Generators in S36
(2 24 28 8 18 34)(3 13 29 9 19 35)(5 25 21 11 31 15)(6 26 22 12 32 16)
(1 23 27 7 17 33)(2 34 18 8 28 24)(3 19 29)(4 36 20 10 30 14)(5 15 31 11 21 25)(6 32 22)(9 13 35)(12 26 16)
(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)

G:=sub<Sym(36)| (2,24,28,8,18,34)(3,13,29,9,19,35)(5,25,21,11,31,15)(6,26,22,12,32,16), (1,23,27,7,17,33)(2,34,18,8,28,24)(3,19,29)(4,36,20,10,30,14)(5,15,31,11,21,25)(6,32,22)(9,13,35)(12,26,16), (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)>;

G:=Group( (2,24,28,8,18,34)(3,13,29,9,19,35)(5,25,21,11,31,15)(6,26,22,12,32,16), (1,23,27,7,17,33)(2,34,18,8,28,24)(3,19,29)(4,36,20,10,30,14)(5,15,31,11,21,25)(6,32,22)(9,13,35)(12,26,16), (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) );

G=PermutationGroup([(2,24,28,8,18,34),(3,13,29,9,19,35),(5,25,21,11,31,15),(6,26,22,12,32,16)], [(1,23,27,7,17,33),(2,34,18,8,28,24),(3,19,29),(4,36,20,10,30,14),(5,15,31,11,21,25),(6,32,22),(9,13,35),(12,26,16)], [(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)])

32 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B···6J6K6L6M6N12A12B12C12D
order1222333333444466···6666612121212
size1133261212242499272726···61212242436363636

32 irreducible representations

dim1111112222333666666
type+++-++++--+-
imageC1C2C3C4C6C12S3Dic3C3×S3C3×Dic3A4C2×A4C4×A4C32⋊C6S3×A4C32⋊C12Dic3×A4C62⋊C6C624C12
kernelC624C12C2×C32⋊A4C22×C3⋊Dic3C32⋊A4C2×C62C62C6×A4C3×A4C22×C6C2×C6C3⋊Dic3C3×C6C32C23C6C22C3C2C1
# reps1122241122112111133

Matrix representation of C624C12 in GL9(𝔽13)

100000000
0120000000
0012000000
000100000
000010000
0000001200
0000011200
0000000121
0000000120
,
1200000000
010000000
0012000000
0000120000
0001120000
0000001200
0000011200
0000000012
0000000112
,
050000000
005000000
500000000
0000012100
000000100
0000000121
000000001
0001210000
000010000

G:=sub<GL(9,GF(13))| [1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12],[0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0] >;

C624C12 in GAP, Magma, Sage, TeX

C_6^2\rtimes_4C_{12}
% in TeX

G:=Group("C6^2:4C12");
// GroupNames label

G:=SmallGroup(432,272);
// by ID

G=gap.SmallGroup(432,272);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,-3,42,514,221,4037,4044,14118]);
// Polycyclic

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

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