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G = C3×C6.D4order 144 = 24·32

Direct product of C3 and C6.D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C6.D4, C624C4, C62.20C22, (C2×C6)⋊4C12, C6.9(C2×C12), (C2×C6)⋊3Dic3, (C3×C6).32D4, (C2×C6).45D6, C6.11(C3×D4), (C6×Dic3)⋊4C2, (C2×Dic3)⋊2C6, (C22×C6).5S3, (C2×C62).2C2, C23.3(C3×S3), C22.7(S3×C6), (C22×C6).8C6, C2.5(C6×Dic3), C6.33(C3⋊D4), C327(C22⋊C4), C6.21(C2×Dic3), C223(C3×Dic3), C32(C3×C22⋊C4), C2.3(C3×C3⋊D4), (C3×C6).30(C2×C4), (C2×C6).10(C2×C6), SmallGroup(144,84)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C6.D4
C1C3C6C2×C6C62C6×Dic3 — C3×C6.D4
C3C6 — C3×C6.D4
C1C2×C6C22×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 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×2], C22 [×2], C6 [×2], C6 [×4], C6 [×11], C2×C4 [×2], C23, C32, Dic3 [×2], C12 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×11], C22⋊C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C22×C6 [×2], C22×C6, C3×Dic3 [×2], C62, C62 [×2], C62 [×2], C6.D4, C3×C22⋊C4, C6×Dic3 [×2], C2×C62, C3×C6.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4 [×2], 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 15 7)(2 24 16 12)(3 23 17 11)(4 22 18 10)(5 21 13 9)(6 20 14 8)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 15 10 18)(8 14 11 17)(9 13 12 16)

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

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

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

G:=TransitiveGroup(24,247);

C3×C6.D4 is a maximal subgroup of
C62.31D4  C62.32D4  C62.94C23  C62.95C23  C62.97C23  C62.98C23  C62.99C23  C62.100C23  C62.101C23  C62.57D4  C623Q8  C62.60D4  C62.111C23  C62.113C23  C62.115C23  C62.116C23  C62.117C23  C625D4  C628D4  C624Q8  C3×S3×C22⋊C4  C12×C3⋊D4  C3×D4×Dic3  C623C12  C62.27D6  C624Dic3  C62.Dic3  C625Dic3  C626Dic3
C3×C6.D4 is a maximal quotient of
C623C12  C62.27D6

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D6A···6F6G···6AE12A···12H
order1222223333344446···66···612···12
size1111221122266661···12···26···6

54 irreducible representations

dim111111112222222222
type+++++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3×S3C3⋊D4C3×D4C3×Dic3S3×C6C3×C3⋊D4
kernelC3×C6.D4C6×Dic3C2×C62C6.D4C62C2×Dic3C22×C6C2×C6C22×C6C3×C6C2×C6C2×C6C23C6C6C22C22C2
# reps121244281221244428

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

300
030
003
,
1200
040
0010
,
500
001
010
,
500
001
0120
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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