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## G = (C6×C12)⋊5C4order 288 = 25·32

### 5th semidirect product of C6×C12 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — (C6×C12)⋊5C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C4×C32⋊C4 — (C6×C12)⋊5C4
 Lower central C32 — C3×C6 — (C6×C12)⋊5C4
 Upper central C1 — C4 — C2×C4

Generators and relations for (C6×C12)⋊5C4
G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a2b >

Subgroups: 656 in 130 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C42⋊C2, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×C32⋊C4, C22×C3⋊S3, C4×C32⋊C4, C4⋊(C32⋊C4), C62⋊C4, C2×C4×C3⋊S3, (C6×C12)⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, (C6×C12)⋊5C4

Permutation representations of (C6×C12)⋊5C4
On 24 points - transitive group 24T621
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 19)(2 20 6 24)(3 21 11 17)(4 22)(5 23 9 15)(7 13)(8 14 12 18)(10 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), (1,19)(2,20,6,24)(3,21,11,17)(4,22)(5,23,9,15)(7,13)(8,14,12,18)(10,16)>;

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

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

G:=TransitiveGroup(24,621);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F ··· 4N 6A ··· 6F 12A ··· 12H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 2 9 9 18 4 4 1 1 2 9 9 18 ··· 18 4 ··· 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4○D4 C32⋊C4 C2×C32⋊C4 C2×C32⋊C4 (C6×C12)⋊5C4 kernel (C6×C12)⋊5C4 C4×C32⋊C4 C4⋊(C32⋊C4) C62⋊C4 C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C6×C12 C3⋊S3 C2×C4 C4 C22 C1 # reps 1 2 2 2 1 4 2 2 4 2 4 2 8

Matrix representation of (C6×C12)⋊5C4 in GL4(𝔽5) generated by

 0 1 1 2 1 3 2 2 0 2 0 4 0 2 4 2
,
 3 3 3 3 4 1 2 0 2 2 3 0 1 3 3 0
,
 0 1 1 2 0 2 4 3 3 1 1 2 1 2 4 2
G:=sub<GL(4,GF(5))| [0,1,0,0,1,3,2,2,1,2,0,4,2,2,4,2],[3,4,2,1,3,1,2,3,3,2,3,3,3,0,0,0],[0,0,3,1,1,2,1,2,1,4,1,4,2,3,2,2] >;

(C6×C12)⋊5C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12})\rtimes_5C_4
% in TeX

G:=Group("(C6xC12):5C4");
// GroupNames label

G:=SmallGroup(288,934);
// by ID

G=gap.SmallGroup(288,934);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,422,9413,362,12550,1203]);
// 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^2*b>;
// generators/relations

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