Copied to
clipboard

G = C3×D46D6order 288 = 25·32

Direct product of C3 and D46D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D46D6, C3272+ 1+4, C62.149C23, (S3×D4)⋊4C6, D46(S3×C6), (C6×D4)⋊7C6, C4○D125C6, (C3×D4)⋊24D6, (C6×D4)⋊16S3, D128(C2×C6), (C2×C12)⋊17D6, C233(S3×C6), (C22×C6)⋊5D6, D42S34C6, Dic68(C2×C6), C6.7(C23×C6), (C6×C12)⋊10C22, (C3×C6).44C24, C6.75(S3×C23), D6.3(C22×C6), (S3×C12)⋊12C22, (C3×D12)⋊34C22, (S3×C6).30C23, C12.21(C22×C6), (C2×C62)⋊10C22, C31(C3×2+ 1+4), C12.172(C22×S3), (C3×C12).122C23, (C6×Dic3)⋊20C22, (C3×Dic6)⋊33C22, (D4×C32)⋊20C22, Dic3.4(C22×C6), (C3×Dic3).31C23, (C2×C4)⋊3(S3×C6), (D4×C3×C6)⋊11C2, (C3×S3×D4)⋊11C2, C4.21(S3×C2×C6), (C4×S3)⋊1(C2×C6), (C2×C12)⋊3(C2×C6), (C3×D4)⋊7(C2×C6), (C2×D4)⋊7(C3×S3), C3⋊D43(C2×C6), C2.8(S3×C22×C6), C22.6(S3×C2×C6), (C2×C3⋊D4)⋊11C6, (C6×C3⋊D4)⋊25C2, (S3×C2×C6)⋊14C22, (C22×C6)⋊6(C2×C6), (C3×C4○D12)⋊15C2, (C22×S3)⋊3(C2×C6), (C2×Dic3)⋊4(C2×C6), (C2×C6).2(C22×C6), (C3×D42S3)⋊11C2, (C3×C3⋊D4)⋊20C22, (C2×C6).21(C22×S3), SmallGroup(288,994)

Series: Derived Chief Lower central Upper central

C1C6 — C3×D46D6
C1C3C6C3×C6S3×C6S3×C2×C6C3×S3×D4 — C3×D46D6
C3C6 — C3×D46D6
C1C6C6×D4

Generators and relations for C3×D46D6
 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 [×9], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], S3 [×4], C6 [×2], C6 [×21], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×8], C2×C6 [×21], C2×D4, C2×D4 [×8], C4○D4 [×6], C3×S3 [×4], C3×C6, C3×C6 [×5], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×9], C3×D4 [×8], C3×D4 [×18], C3×Q8 [×2], C22×S3 [×4], C22×C6 [×4], C22×C6 [×6], 2+ 1+4, C3×Dic3 [×4], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C62 [×4], C62 [×2], C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4 [×2], C6×D4 [×9], C3×C4○D4 [×6], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C6×Dic3 [×4], C3×C3⋊D4 [×12], C6×C12, D4×C32 [×4], S3×C2×C6 [×4], C2×C62 [×2], D46D6, C3×2+ 1+4, C3×C4○D12 [×2], C3×S3×D4 [×4], C3×D42S3 [×4], C6×C3⋊D4 [×4], D4×C3×C6, C3×D46D6
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], 2+ 1+4, S3×C6 [×7], S3×C23, C23×C6, S3×C2×C6 [×7], D46D6, C3×2+ 1+4, S3×C22×C6, C3×D46D6

Permutation representations of C3×D46D6
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+++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C3×S3S3×C6S3×C6S3×C62+ 1+4D46D6C3×2+ 1+4C3×D46D6
kernelC3×D46D6C3×C4○D12C3×S3×D4C3×D42S3C6×C3⋊D4D4×C3×C6D46D6C4○D12S3×D4D42S3C2×C3⋊D4C6×D4C6×D4C2×C12C3×D4C22×C6C2×D4C2×C4D4C23C32C3C3C1
# reps124441248882114222841224

Matrix representation of C3×D46D6 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] >;

C3×D46D6 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

׿
×
𝔽