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## 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 [×4], C3 [×2], C4 [×3], C22, C22 [×5], S3 [×8], C6 [×6], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C32, C12 [×4], D6 [×16], C2×C6 [×2], C22⋊C4 [×2], C2×D4, C3⋊S3 [×3], C3×C6, C3×C6, D12 [×8], C2×C12 [×2], C22×S3 [×4], C23⋊C4, C3×C12, C32⋊C4 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×3], C62, C2×D12 [×2], C12⋊S3 [×2], C6×C12, C2×C32⋊C4 [×2], C22×C3⋊S3 [×2], C62⋊C4 [×2], C2×C12⋊S3, (C6×C12)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 16)(2 23 12 21)(3 18 11 14)(4 13 10 19)(5 20 9 24)(6 15 8 17)(7 22)

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

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

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

G:=TransitiveGroup(24,622);

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

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

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

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

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

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