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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4.D4
 Lower central C1 — C2 — C22 — C3×C4.D4
 Upper central C1 — C6 — C2×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 >

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

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

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

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

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 2A 2B 2C 2D 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 2 3 3 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 2 4 4 1 1 2 2 1 1 2 2 4 4 4 4 4 4 4 4 2 2 2 2 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 C4.D4 C3×C4.D4 kernel C3×C4.D4 C3×M4(2) C6×D4 C4.D4 C22×C6 M4(2) C2×D4 C23 C12 C4 C3 C1 # reps 1 2 1 2 4 4 2 8 2 4 1 2

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

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 5 4 6 6 0 6 0 3 3 3 1 3 4 2 0
,
 0 2 5 2 0 0 5 6 5 2 1 3 4 4 3 6
,
 4 1 0 4 4 3 3 0 2 5 6 4 3 3 2 1
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

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