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G = C2xC6.D6order 144 = 24·32

Direct product of C2 and C6.D6

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

Aliases: C2xC6.D6, Dic3:6D6, C62.9C22, C6:1(C4xS3), C22.9S32, (C2xC6).14D6, (C6xDic3):8C2, (C2xDic3):5S3, C32:4(C22xC4), (C3xC6).13C23, C6.13(C22xS3), (C3xDic3):7C22, C3:2(S3xC2xC4), C2.3(C2xS32), (C2xC3:S3):4C4, C3:S3:2(C2xC4), (C3xC6):3(C2xC4), (C22xC3:S3).4C2, (C2xC3:S3).16C22, SmallGroup(144,149)

Series: Derived Chief Lower central Upper central

C1C32 — C2xC6.D6
C1C3C32C3xC6C3xDic3C6.D6 — C2xC6.D6
C32 — C2xC6.D6
C1C22

Generators and relations for C2xC6.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, C2xC4, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22xC4, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C3xDic3, C2xC3:S3, C62, S3xC2xC4, C6.D6, C6xDic3, C22xC3:S3, C2xC6.D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S32, S3xC2xC4, C6.D6, C2xS32, C2xC6.D6

Permutation representations of C2xC6.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);

C2xC6.D6 is a maximal subgroup of
C62.D4  C62.6C23  C62.24C23  Dic3:4D12  C62.53C23  C62.58C23  Dic3:5D12  C62.65C23  C62.67C23  C62.70C23  C62.74C23  Dic3:3D12  C62.91C23  C62.94C23  C62.100C23  C62.116C23  C62.117C23  C62.9D4  S32xC2xC4  Dic6:12D6
C2xC6.D6 is a maximal quotient of
C3:C8.22D6  C3:C8:20D6  Dic3:6Dic6  C62.19C23  C62.44C23  Dic3:5D12  C62.70C23  C62.94C23  C2xDic32  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++++++++++
imageC1C2C2C2C4S3D6D6C4xS3S32C6.D6C2xS32
kernelC2xC6.D6C6.D6C6xDic3C22xC3:S3C2xC3:S3C2xDic3Dic3C2xC6C6C22C2C2
# reps142182428121

Matrix representation of C2xC6.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] >;

C2xC6.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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