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## G = C2×C6.D6order 144 = 24·32

### Direct product of C2 and C6.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C6.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6 — C2×C6.D6
 Lower central C32 — C2×C6.D6
 Upper central C1 — C22

Generators and relations for C2×C6.D6
G = < a,b,c,d | a2=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 400 in 124 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C3×Dic3, C2×C3⋊S3, C62, S3×C2×C4, C6.D6, C6×Dic3, C22×C3⋊S3, C2×C6.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C6.D6, C2×S32, C2×C6.D6

Permutation representations of C2×C6.D6
On 24 points - transitive group 24T225
Generators in S24
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(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 9)(3 7)(4 12)(6 10)(13 21)(15 19)(16 24)(18 22)

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

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

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

G:=TransitiveGroup(24,225);

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D6 C4×S3 S32 C6.D6 C2×S32 kernel C2×C6.D6 C6.D6 C6×Dic3 C22×C3⋊S3 C2×C3⋊S3 C2×Dic3 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 2 1 8 2 4 2 8 1 2 1

Matrix representation of C2×C6.D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C6.D6 in GAP, Magma, Sage, TeX

C_2\times C_6.D_6
% in TeX

G:=Group("C2xC6.D6");
// GroupNames label

G:=SmallGroup(144,149);
// by ID

G=gap.SmallGroup(144,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,55,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=d^2=1,c^6=b^3,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^5>;
// generators/relations

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