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G = C2×C6.D6order 144 = 24·32

Direct product of C2 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C6.D6, Dic36D6, C62.9C22, C61(C4×S3), C22.9S32, (C2×C6).14D6, (C6×Dic3)⋊8C2, (C2×Dic3)⋊5S3, C324(C22×C4), (C3×C6).13C23, C6.13(C22×S3), (C3×Dic3)⋊7C22, C32(S3×C2×C4), C2.3(C2×S32), (C2×C3⋊S3)⋊4C4, C3⋊S32(C2×C4), (C3×C6)⋊3(C2×C4), (C22×C3⋊S3).4C2, (C2×C3⋊S3).16C22, SmallGroup(144,149)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C6.D6
C1C3C32C3×C6C3×Dic3C6.D6 — C2×C6.D6
C32 — C2×C6.D6
C1C22

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 [×2], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4 [×6], C23, C32, Dic3 [×4], C12 [×4], D6 [×18], C2×C6 [×2], C2×C6, C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×3], C3×Dic3 [×4], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×2], C6.D6 [×4], C6×Dic3 [×2], C22×C3⋊S3, C2×C6.D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], S32, S3×C2×C4 [×2], C6.D6 [×2], 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);

C2×C6.D6 is a maximal subgroup of
C62.D4  C62.6C23  C62.24C23  Dic34D12  C62.53C23  C62.58C23  Dic35D12  C62.65C23  C62.67C23  C62.70C23  C62.74C23  Dic33D12  C62.91C23  C62.94C23  C62.100C23  C62.116C23  C62.117C23  C62.9D4  S32×C2×C4  Dic612D6
C2×C6.D6 is a maximal quotient of
C3⋊C8.22D6  C3⋊C820D6  Dic36Dic6  C62.19C23  C62.44C23  Dic35D12  C62.70C23  C62.94C23  C2×Dic32  C62.99C23  C62.116C23

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A···4H6A···6F6G6H6I12A···12H
order122222223334···46···666612···12
size111199992243···32···24446···6

36 irreducible representations

dim111112222444
type++++++++++
imageC1C2C2C2C4S3D6D6C4×S3S32C6.D6C2×S32
kernelC2×C6.D6C6.D6C6×Dic3C22×C3⋊S3C2×C3⋊S3C2×Dic3Dic3C2×C6C6C22C2C2
# reps142182428121

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

1200000
0120000
0012000
0001200
000010
000001
,
0120000
1120000
000100
0012100
000010
000001
,
010000
100000
000800
008000
000001
00001212
,
0120000
1200000
000100
001000
000001
000010

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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