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G = C3×C12.D4order 288 = 25·32

Direct product of C3 and C12.D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.D4, (C6×D4).2C6, C12.8(C3×D4), (C6×D4).23S3, (C3×C12).46D4, C4.Dic33C6, (C2×C62).2C4, (C2×C12).222D6, (C22×C6).5C12, C62.97(C2×C4), (C6×C12).48C22, C327(C4.D4), (C22×C6).5Dic3, C22.2(C6×Dic3), C23.2(C3×Dic3), C12.100(C3⋊D4), C6.32(C6.D4), (D4×C3×C6).2C2, (C2×C4).3(S3×C6), (C2×D4).2(C3×S3), C32(C3×C4.D4), C4.13(C3×C3⋊D4), (C2×C6).38(C2×C12), (C2×C12).18(C2×C6), C6.14(C3×C22⋊C4), (C3×C4.Dic3)⋊19C2, (C2×C6).23(C2×Dic3), C2.4(C3×C6.D4), (C3×C6).65(C22⋊C4), SmallGroup(288,267)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C12.D4
C1C3C6C2×C6C2×C12C6×C12C3×C4.Dic3 — C3×C12.D4
C3C6C2×C6 — C3×C12.D4
C1C6C2×C12C6×D4

Generators and relations for C3×C12.D4
 G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >

Subgroups: 266 in 119 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], C6 [×2], C6 [×13], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6 [×17], M4(2) [×2], C2×D4, C3×C6, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×C12 [×2], C2×C12, C3×D4 [×8], C22×C6 [×4], C22×C6 [×2], C4.D4, C3×C12 [×2], C62, C62 [×4], C4.Dic3 [×2], C3×M4(2) [×2], C6×D4 [×2], C6×D4, C3×C3⋊C8 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C12.D4, C3×C4.D4, C3×C4.Dic3 [×2], D4×C3×C6, C3×C12.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], C4.D4, C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4 [×2], C12.D4, C3×C4.D4, C3×C6.D4, C3×C12.D4

Permutation representations of C3×C12.D4
On 24 points - transitive group 24T590
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 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 17 10 20 7 23 4 14)(2 16 11 19 8 22 5 13)(3 15 12 18 9 21 6 24)
(1 20 10 17 7 14 4 23)(2 13 11 22 8 19 5 16)(3 18 12 15 9 24 6 21)

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

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

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

G:=TransitiveGroup(24,590);

63 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B6A6B6C···6M6N···6AC8A8B8C8D12A12B12C12D12E···12J24A···24H
order122223333344666···66···688881212121212···1224···24
size112441122222112···24···41212121222224···412···12

63 irreducible representations

dim1111111122222222224444
type++++++-+
imageC1C2C2C3C4C6C6C12S3D4D6Dic3C3×S3C3⋊D4C3×D4S3×C6C3×Dic3C3×C3⋊D4C4.D4C12.D4C3×C4.D4C3×C12.D4
kernelC3×C12.D4C3×C4.Dic3D4×C3×C6C12.D4C2×C62C4.Dic3C6×D4C22×C6C6×D4C3×C12C2×C12C22×C6C2×D4C12C12C2×C4C23C4C32C3C3C1
# reps1212442812122442481224

Matrix representation of C3×C12.D4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
3102
0543
6662
1630
,
6435
6630
2566
2253
,
4030
4505
5211
5514
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,6,1,1,5,6,6,0,4,6,3,2,3,2,0],[6,6,2,2,4,6,5,2,3,3,6,5,5,0,6,3],[4,4,5,5,0,5,2,5,3,0,1,1,0,5,1,4] >;

C3×C12.D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}.D_4
% in TeX

G:=Group("C3xC12.D4");
// GroupNames label

G:=SmallGroup(288,267);
// by ID

G=gap.SmallGroup(288,267);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,850,136,2524,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

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