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G = C3×C4.D4order 96 = 25·3

Direct product of C3 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×C4.D4, C23.C12, C12.58D4, M4(2)⋊3C6, C4.9(C3×D4), (C2×D4).2C6, (C6×D4).8C2, (C22×C6).1C4, (C3×M4(2))⋊9C2, C22.3(C2×C12), C6.22(C22⋊C4), (C2×C12).59C22, (C2×C4).1(C2×C6), (C2×C6).20(C2×C4), C2.4(C3×C22⋊C4), SmallGroup(96,50)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C4.D4
C1C2C4C2×C4C2×C12C3×M4(2) — C3×C4.D4
C1C2C22 — C3×C4.D4
C1C6C2×C12 — C3×C4.D4

Generators and relations for C3×C4.D4
 G = < a,b,c,d | a3=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

2C2
4C2
4C2
2C22
2C22
4C22
4C22
2C6
4C6
4C6
2D4
2C8
2D4
2C8
2C2×C6
2C2×C6
4C2×C6
4C2×C6
2C3×D4
2C24
2C24
2C3×D4

Permutation representations of C3×C4.D4
On 24 points - transitive group 24T90
Generators in S24
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 18)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)
(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 6 3 4 5 2 7 8)(9 10 15 12 13 14 11 16)(17 18 19 24 21 22 23 20)

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

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

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

G:=TransitiveGroup(24,90);

C3×C4.D4 is a maximal subgroup of   C23.3D12  C23.4D12  M4(2).19D6  M4(2)⋊D6  D121D4  D12.2D4  D12.3D4

33 conjugacy classes

class 1 2A2B2C2D3A3B4A4B6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C12D24A···24H
order1222233446666666688881212121224···24
size11244112211224444444422224···4

33 irreducible representations

dim111111112244
type+++++
imageC1C2C2C3C4C6C6C12D4C3×D4C4.D4C3×C4.D4
kernelC3×C4.D4C3×M4(2)C6×D4C4.D4C22×C6M4(2)C2×D4C23C12C4C3C1
# reps121244282412

Matrix representation of C3×C4.D4 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
4546
6060
3331
3420
,
0252
0056
5213
4436
,
4104
4330
2564
3321
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0],[0,0,5,4,2,0,2,4,5,5,1,3,2,6,3,6],[4,4,2,3,1,3,5,3,0,3,6,2,4,0,4,1] >;

C3×C4.D4 in GAP, Magma, Sage, TeX

C_3\times C_4.D_4
% in TeX

G:=Group("C3xC4.D4");
// GroupNames label

G:=SmallGroup(96,50);
// by ID

G=gap.SmallGroup(96,50);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,1090,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4.D4 in TeX

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