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G = C3×S3×C3⋊D4order 432 = 24·33

Direct product of C3, S3 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×C3⋊D4, C6229D6, D66(S3×C6), (S3×C6)⋊17D6, C629(C2×C6), C329(C6×D4), C3319(C2×D4), C3⋊D125C6, D6⋊S35C6, (S3×C62)⋊3C2, (S3×Dic3)⋊3C6, Dic31(S3×C6), (S3×C32)⋊6D4, C3226(S3×D4), C327D46C6, (C3×Dic3)⋊10D6, (C3×C62)⋊1C22, (C32×C6).36C23, (C32×Dic3)⋊5C22, (C2×C6)⋊8S32, C35(C3×S3×D4), (C2×S32)⋊3C6, (S32×C6)⋊7C2, (S3×C2×C6)⋊5C6, (S3×C2×C6)⋊5S3, C2.17(S32×C6), C223(C3×S32), C32(C6×C3⋊D4), C6.17(S3×C2×C6), (S3×C6)⋊6(C2×C6), C6.120(C2×S32), (C2×C6)⋊10(S3×C6), (C3×S3)⋊2(C3×D4), (C3×C3⋊D4)⋊3C6, (S3×C3×C6)⋊9C22, (C3×S3×Dic3)⋊8C2, (C6×C3⋊S3)⋊7C22, C3⋊Dic33(C2×C6), (C22×S3)⋊5(C3×S3), (C3×Dic3)⋊1(C2×C6), (C3×C327D4)⋊8C2, (C32×C3⋊D4)⋊3C2, C3218(C2×C3⋊D4), (C3×C3⋊D12)⋊11C2, (C3×D6⋊S3)⋊12C2, (C3×C6).27(C22×C6), (C3×C6).141(C22×S3), (C3×C3⋊Dic3)⋊10C22, (C2×C3⋊S3)⋊5(C2×C6), SmallGroup(432,658)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×S3×C3⋊D4
C1C3C32C3×C6C32×C6S3×C3×C6S32×C6 — C3×S3×C3⋊D4
C32C3×C6 — C3×S3×C3⋊D4
C1C6C2×C6

Generators and relations for C3×S3×C3⋊D4
 G = < a,b,c,d,e,f | a3=b3=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1032 in 290 conjugacy classes, 72 normal (64 characteristic)
C1, C2, C2 [×6], C3 [×3], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×2], S3 [×5], C6 [×3], C6 [×27], C2×C4, D4 [×4], C23 [×2], C32 [×3], C32 [×4], Dic3, Dic3 [×3], C12 [×4], D6 [×3], D6 [×9], C2×C6 [×3], C2×C6 [×24], C2×D4, C3×S3 [×4], C3×S3 [×12], C3⋊S3, C3×C6 [×3], C3×C6 [×17], C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4 [×6], C2×C12, C3×D4 [×6], C22×S3, C22×S3 [×2], C22×C6 [×4], C33, C3×Dic3 [×2], C3×Dic3 [×4], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×6], S3×C6 [×18], C2×C3⋊S3, C62 [×3], C62 [×9], S3×D4, C2×C3⋊D4, C6×D4, S3×C32 [×2], S3×C32 [×2], C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4 [×2], C3×C3⋊D4 [×7], C327D4, D4×C32, C2×S32, S3×C2×C6 [×2], S3×C2×C6 [×3], C2×C62, C32×Dic3, C3×C3⋊Dic3, C3×S32 [×2], S3×C3×C6 [×3], S3×C3×C6 [×2], C6×C3⋊S3, C3×C62, S3×C3⋊D4, C3×S3×D4, C6×C3⋊D4, C3×S3×Dic3, C3×D6⋊S3, C3×C3⋊D12, C32×C3⋊D4, C3×C327D4, S32×C6, S3×C62, C3×S3×C3⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×2], C6 [×7], D4 [×2], C23, D6 [×6], C2×C6 [×7], C2×D4, C3×S3 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C22×C6, S32, S3×C6 [×6], S3×D4, C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], C2×S32, S3×C2×C6 [×2], C3×S32, S3×C3⋊D4, C3×S3×D4, C6×C3⋊D4, S32×C6, C3×S3×C3⋊D4

Permutation representations of C3×S3×C3⋊D4
On 24 points - transitive group 24T1280
Generators in S24
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 10 23)(6 24 11)(7 12 21)(8 22 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)

G:=sub<Sym(24)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;

G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24) );

G=PermutationGroup([(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,17,16),(2,13,18),(3,19,14),(4,15,20),(5,10,23),(6,24,11),(7,12,21),(8,22,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24)])

G:=TransitiveGroup(24,1280);

81 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H3I3J3K4A4B6A6B6C···6P6Q6R6S6T6U···6AF6AG···6AV6AW6AX6AY6AZ6BA12A12B12C12D12E12F12G
order12222222333···333344666···666666···66···66666612121212121212
size112336618112···2444618112···233334···46···61212121818661212121818

81 irreducible representations

dim11111111111111112222222222222244444444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3S3D4D6D6D6C3×S3C3×S3C3⋊D4C3×D4S3×C6S3×C6S3×C6C3×C3⋊D4S32S3×D4C2×S32C3×S32S3×C3⋊D4C3×S3×D4S32×C6C3×S3×C3⋊D4
kernelC3×S3×C3⋊D4C3×S3×Dic3C3×D6⋊S3C3×C3⋊D12C32×C3⋊D4C3×C327D4S32×C6S3×C62S3×C3⋊D4S3×Dic3D6⋊S3C3⋊D12C3×C3⋊D4C327D4C2×S32S3×C2×C6C3×C3⋊D4S3×C2×C6S3×C32C3×Dic3S3×C6C62C3⋊D4C22×S3C3×S3C3×S3Dic3D6C2×C6S3C2×C6C32C6C22C3C3C2C1
# reps11111111222222221121322244264811122224

Matrix representation of C3×S3×C3⋊D4 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
6031
5360
1645
3326
,
3315
2520
6612
3155
,
4121
3364
6642
6141
,
3100
4400
4563
0341
,
6350
2446
3102
4454
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[3,2,6,3,3,5,6,1,1,2,1,5,5,0,2,5],[4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[6,2,3,4,3,4,1,4,5,4,0,5,0,6,2,4] >;

C3×S3×C3⋊D4 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_3\rtimes D_4
% in TeX

G:=Group("C3xS3xC3:D4");
// GroupNames label

G:=SmallGroup(432,658);
// by ID

G=gap.SmallGroup(432,658);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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