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G = C6×C3⋊D4order 144 = 24·32

Direct product of C6 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C6×C3⋊D4, C627C22, (C2×C6)⋊9D6, C33(C6×D4), (C3×C6)⋊6D4, C62(C3×D4), D63(C2×C6), C224(S3×C6), (C2×C62)⋊2C2, C233(C3×S3), (C22×C6)⋊4C6, (C22×C6)⋊3S3, C3212(C2×D4), (C22×S3)⋊4C6, (S3×C6)⋊9C22, (C2×Dic3)⋊4C6, Dic32(C2×C6), (C6×Dic3)⋊10C2, C6.10(C22×C6), (C3×C6).28C23, C6.49(C22×S3), (C3×Dic3)⋊9C22, (S3×C2×C6)⋊6C2, (C2×C6)⋊5(C2×C6), C2.10(S3×C2×C6), SmallGroup(144,167)

Series: Derived Chief Lower central Upper central

C1C6 — C6×C3⋊D4
C1C3C6C3×C6S3×C6S3×C2×C6 — C6×C3⋊D4
C3C6 — C6×C3⋊D4
C1C2×C6C22×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 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×2], C6 [×4], C6 [×13], C2×C4, D4 [×4], C23, C23, C32, Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×15], C2×D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C22×C6 [×2], C3×Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×2], C62 [×2], C2×C3⋊D4, C6×D4, C6×Dic3, C3×C3⋊D4 [×4], S3×C2×C6, C2×C62, C6×C3⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], 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 13 11)(2 21 14 12)(3 22 15 7)(4 23 16 8)(5 24 17 9)(6 19 18 10)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)

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

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

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

G:=TransitiveGroup(24,248);

C6×C3⋊D4 is a maximal subgroup of
C62.31D4  C62.100C23  C62.101C23  C62.56D4  C62.57D4  C62.111C23  C62.112C23  C62.113C23  C62.115C23  C624D4  C626D4  C62.121C23  C627D4  C628D4  C62.125C23  C32⋊2+ 1+4  S3×C6×D4

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A···6F6G···6AE6AF6AG6AH6AI12A12B12C12D
order1222222233333446···66···6666612121212
size1111226611222661···12···266666666

54 irreducible representations

dim111111111122222222
type++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4
kernelC6×C3⋊D4C6×Dic3C3×C3⋊D4S3×C2×C6C2×C62C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C22×C6C3×C6C2×C6C23C6C6C22C2
# reps114112282212324468

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

400
0100
0010
,
100
030
009
,
100
001
0120
,
100
001
010
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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