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## G = C6×C3⋊D4order 144 = 24·32

### Direct product of C6 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×C3⋊D4
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C6×C3⋊D4
 Lower central C3 — C6 — C6×C3⋊D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C3⋊D4
G = < a,b,c,d | a6=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 248 in 124 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×Dic3, S3×C6, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C6×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C6×C3⋊D4

Permutation representations of C6×C3⋊D4
On 24 points - transitive group 24T248
Generators in S24
(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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 20 15 11)(2 21 16 12)(3 22 17 7)(4 23 18 8)(5 24 13 9)(6 19 14 10)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)

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

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

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

G:=TransitiveGroup(24,248);

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6AE 6AF 6AG 6AH 6AI 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 2 2 6 6 1 1 2 2 2 6 6 1 ··· 1 2 ··· 2 6 6 6 6 6 6 6 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C6×C3⋊D4 C6×Dic3 C3×C3⋊D4 S3×C2×C6 C2×C62 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C22×C6 C3×C6 C2×C6 C23 C6 C6 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 2 3 2 4 4 6 8

Matrix representation of C6×C3⋊D4 in GL3(𝔽13) generated by

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

C6×C3⋊D4 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes D_4
% in TeX

G:=Group("C6xC3:D4");
// GroupNames label

G:=SmallGroup(144,167);
// by ID

G=gap.SmallGroup(144,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,506,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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