Copied to
clipboard

## G = (C6×C12)⋊C4order 288 = 25·32

### 1st semidirect product of C6×C12 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — (C6×C12)⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C62⋊C4 — (C6×C12)⋊C4
 Lower central C32 — C3×C6 — C62 — (C6×C12)⋊C4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for (C6×C12)⋊C4
G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a-1b7 >

Subgroups: 792 in 106 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C32, C12, D6, C2×C6, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6, D12, C2×C12, C22×S3, C23⋊C4, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C12⋊S3, C6×C12, C2×C32⋊C4, C22×C3⋊S3, C62⋊C4, C2×C12⋊S3, (C6×C12)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C23⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C6×C12)⋊C4

Character table of (C6×C12)⋊C4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 18 18 36 4 4 4 36 36 36 36 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 1 1 1 -1 -i -i i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 -1 -1 -1 1 1 1 i -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 1 -i i -i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 1 1 1 -1 i i -i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 2 -2 0 2 2 0 0 0 0 0 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 2 2 0 0 0 0 0 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 -2 1 0 0 0 0 0 -1 -1 2 1 -2 2 3 0 -3 0 0 3 -3 0 orthogonal lifted from C62⋊C4 ρ12 4 4 4 0 0 0 -2 1 -4 0 0 0 0 1 1 -2 1 -2 -2 -1 2 -1 2 2 -1 -1 2 orthogonal lifted from C2×C32⋊C4 ρ13 4 -4 0 0 0 0 4 4 0 0 0 0 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ14 4 4 -4 0 0 0 1 -2 0 0 0 0 0 2 2 -1 -2 1 -1 0 3 0 -3 -3 0 0 3 orthogonal lifted from C62⋊C4 ρ15 4 4 -4 0 0 0 1 -2 0 0 0 0 0 2 2 -1 -2 1 -1 0 -3 0 3 3 0 0 -3 orthogonal lifted from C62⋊C4 ρ16 4 4 -4 0 0 0 -2 1 0 0 0 0 0 -1 -1 2 1 -2 2 -3 0 3 0 0 -3 3 0 orthogonal lifted from C62⋊C4 ρ17 4 4 4 0 0 0 -2 1 4 0 0 0 0 1 1 -2 1 -2 -2 1 -2 1 -2 -2 1 1 -2 orthogonal lifted from C32⋊C4 ρ18 4 4 4 0 0 0 1 -2 -4 0 0 0 0 -2 -2 1 -2 1 1 2 -1 2 -1 -1 2 2 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 4 0 0 0 1 -2 4 0 0 0 0 -2 -2 1 -2 1 1 -2 1 -2 1 1 -2 -2 1 orthogonal lifted from C32⋊C4 ρ20 4 -4 0 0 0 0 -2 1 0 0 0 0 0 3 -3 0 -1 2 0 -√3 2√3 -√3 0 0 √3 √3 -2√3 orthogonal faithful ρ21 4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 -3 2 -1 3 0 √3 2√3 √3 -√3 0 -2√3 -√3 orthogonal faithful ρ22 4 -4 0 0 0 0 -2 1 0 0 0 0 0 -3 3 0 -1 2 0 -√3 0 √3 -2√3 2√3 √3 -√3 0 orthogonal faithful ρ23 4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 -3 2 -1 3 0 -√3 -2√3 -√3 √3 0 2√3 √3 orthogonal faithful ρ24 4 -4 0 0 0 0 -2 1 0 0 0 0 0 -3 3 0 -1 2 0 √3 0 -√3 2√3 -2√3 -√3 √3 0 orthogonal faithful ρ25 4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 3 2 -1 -3 -2√3 -√3 0 √3 -√3 2√3 0 √3 orthogonal faithful ρ26 4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 3 2 -1 -3 2√3 √3 0 -√3 √3 -2√3 0 -√3 orthogonal faithful ρ27 4 -4 0 0 0 0 -2 1 0 0 0 0 0 3 -3 0 -1 2 0 √3 -2√3 √3 0 0 -√3 -√3 2√3 orthogonal faithful

Permutation representations of (C6×C12)⋊C4
On 24 points - transitive group 24T622
Generators in S24
(13 15 17 19 21 23)(14 16 18 20 22 24)
(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 22)(2 17 12 15)(3 24 11 20)(4 19 10 13)(5 14 9 18)(6 21 8 23)(7 16)

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

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

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

G:=TransitiveGroup(24,622);

On 24 points - transitive group 24T623
Generators in S24
(1 10 7 3 12 5)(2 11 8 4 9 6)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(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 18 4 21)(2 15 3 24)(5 16 9 23)(6 13 12 14)(7 22 11 17)(8 19 10 20)

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

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

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

G:=TransitiveGroup(24,623);

Matrix representation of (C6×C12)⋊C4 in GL4(𝔽13) generated by

 0 12 0 0 1 12 0 0 11 8 12 0 11 8 0 12
,
 10 10 0 0 3 7 0 0 9 3 7 10 3 1 3 10
,
 0 0 12 1 2 5 11 12 9 7 9 12 2 4 9 12
G:=sub<GL(4,GF(13))| [0,1,11,11,12,12,8,8,0,0,12,0,0,0,0,12],[10,3,9,3,10,7,3,1,0,0,7,3,0,0,10,10],[0,2,9,2,0,5,7,4,12,11,9,9,1,12,12,12] >;

(C6×C12)⋊C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12})\rtimes C_4
% in TeX

G:=Group("(C6xC12):C4");
// GroupNames label

G:=SmallGroup(288,422);
// by ID

G=gap.SmallGroup(288,422);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,219,675,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^6=b^12=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^-1*b^7>;
// generators/relations

Export

׿
×
𝔽