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G = C3xD4:6D6order 288 = 25·32

Direct product of C3 and D4:6D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xD4:6D6, C32:72+ 1+4, C62.149C23, (S3xD4):4C6, D4:6(S3xC6), (C6xD4):7C6, C4oD12:5C6, (C3xD4):24D6, (C6xD4):16S3, D12:8(C2xC6), (C2xC12):17D6, C23:3(S3xC6), (C22xC6):5D6, D4:2S3:4C6, Dic6:8(C2xC6), C6.7(C23xC6), (C6xC12):10C22, (C3xC6).44C24, C6.75(S3xC23), D6.3(C22xC6), (S3xC12):12C22, (C3xD12):34C22, (S3xC6).30C23, C12.21(C22xC6), (C2xC62):10C22, C3:1(C3x2+ 1+4), C12.172(C22xS3), (C3xC12).122C23, (C6xDic3):20C22, (C3xDic6):33C22, (D4xC32):20C22, Dic3.4(C22xC6), (C3xDic3).31C23, (C2xC4):3(S3xC6), (D4xC3xC6):11C2, (C3xS3xD4):11C2, C4.21(S3xC2xC6), (C4xS3):1(C2xC6), (C2xC12):3(C2xC6), (C3xD4):7(C2xC6), (C2xD4):7(C3xS3), C3:D4:3(C2xC6), C2.8(S3xC22xC6), C22.6(S3xC2xC6), (C2xC3:D4):11C6, (C6xC3:D4):25C2, (S3xC2xC6):14C22, (C22xC6):6(C2xC6), (C3xC4oD12):15C2, (C22xS3):3(C2xC6), (C2xDic3):4(C2xC6), (C2xC6).2(C22xC6), (C3xD4:2S3):11C2, (C3xC3:D4):20C22, (C2xC6).21(C22xS3), SmallGroup(288,994)

Series: Derived Chief Lower central Upper central

C1C6 — C3xD4:6D6
C1C3C6C3xC6S3xC6S3xC2xC6C3xS3xD4 — C3xD4:6D6
C3C6 — C3xD4:6D6
C1C6C6xD4

Generators and relations for C3xD4:6D6
 G = < a,b,c,d,e | a3=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 810 in 359 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C2xD4, C2xD4, C4oD4, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C22xC6, C22xC6, 2+ 1+4, C3xDic3, C3xC12, S3xC6, S3xC6, C62, C62, C62, C4oD12, S3xD4, D4:2S3, C2xC3:D4, C6xD4, C6xD4, C3xC4oD4, C3xDic6, S3xC12, C3xD12, C6xDic3, C3xC3:D4, C6xC12, D4xC32, S3xC2xC6, C2xC62, D4:6D6, C3x2+ 1+4, C3xC4oD12, C3xS3xD4, C3xD4:2S3, C6xC3:D4, D4xC3xC6, C3xD4:6D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C24, C3xS3, C22xS3, C22xC6, 2+ 1+4, S3xC6, S3xC23, C23xC6, S3xC2xC6, D4:6D6, C3x2+ 1+4, S3xC22xC6, C3xD4:6D6

Permutation representations of C3xD4:6D6
On 24 points - transitive group 24T588
Generators in S24
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 4 15 18)(2 13 16 5)(3 6 17 14)(7 10 19 22)(8 23 20 11)(9 12 21 24)
(2 16)(4 18)(6 14)(7 19)(9 21)(11 23)
(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 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,588);

81 conjugacy classes

class 1 2A2B···2F2G2H2I2J3A3B3C3D3E4A4B4C4D4E4F6A6B6C···6U6V···6AG6AH···6AO12A12B12C12D12E···12J12K···12R
order122···2222233333444444666···66···66···61212121212···1212···12
size112···2666611222226666112···24···46···622224···46···6

81 irreducible representations

dim111111111111222222224444
type+++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C3xS3S3xC6S3xC6S3xC62+ 1+4D4:6D6C3x2+ 1+4C3xD4:6D6
kernelC3xD4:6D6C3xC4oD12C3xS3xD4C3xD4:2S3C6xC3:D4D4xC3xC6D4:6D6C4oD12S3xD4D4:2S3C2xC3:D4C6xD4C6xD4C2xC12C3xD4C22xC6C2xD4C2xC4D4C23C32C3C3C1
# reps124441248882114222841224

Matrix representation of C3xD4:6D6 in GL4(F7) generated by

2000
0200
0020
0002
,
0064
3063
4426
6145
,
6160
6623
6533
0436
,
4415
5465
6463
4450
,
3350
6046
6132
0001
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,4,6,0,0,4,1,6,6,2,4,4,3,6,5],[6,6,6,0,1,6,5,4,6,2,3,3,0,3,3,6],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;

C3xD4:6D6 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_6D_6
% in TeX

G:=Group("C3xD4:6D6");
// GroupNames label

G:=SmallGroup(288,994);
// by ID

G=gap.SmallGroup(288,994);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,555,1571,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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