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## G = C3×C6.D4order 144 = 24·32

### Direct product of C3 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C3×C6.D4
 Lower central C3 — C6 — C3×C6.D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C3×C6.D4
G = < a,b,c,d | a3=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 152 in 84 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C6×Dic3, C2×C62, C3×C6.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, C3×C6.D4

Permutation representations of C3×C6.D4
On 24 points - transitive group 24T247
Generators in S24
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(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 17 7)(2 24 18 12)(3 23 13 11)(4 22 14 10)(5 21 15 9)(6 20 16 8)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 17 10 14)(8 16 11 13)(9 15 12 18)

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

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

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

G:=TransitiveGroup(24,247);

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 C3×C3⋊D4 kernel C3×C6.D4 C6×Dic3 C2×C62 C6.D4 C62 C2×Dic3 C22×C6 C2×C6 C22×C6 C3×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 # reps 1 2 1 2 4 4 2 8 1 2 2 1 2 4 4 4 2 8

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

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

C3×C6.D4 in GAP, Magma, Sage, TeX

C_3\times C_6.D_4
% in TeX

G:=Group("C3xC6.D4");
// GroupNames label

G:=SmallGroup(144,84);
// by ID

G=gap.SmallGroup(144,84);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,3461]);
// Polycyclic

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

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