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## G = C3×C12.D4order 288 = 25·32

### Direct product of C3 and C12.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C3×C4.Dic3 — C3×C12.D4
 Lower central C3 — C6 — C2×C6 — C3×C12.D4
 Upper central C1 — C6 — C2×C12 — C6×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, C3, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, C2×C6, C2×C6, M4(2), C2×D4, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C4.D4, C3×C12, C62, C62, C4.Dic3, C3×M4(2), C6×D4, C6×D4, C3×C3⋊C8, C6×C12, D4×C32, C2×C62, C12.D4, C3×C4.D4, C3×C4.Dic3, D4×C3×C6, C3×C12.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C4.D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, 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 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C ··· 6M 6N ··· 6AC 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 3 3 3 3 3 4 4 6 6 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 4 4 1 1 2 2 2 2 2 1 1 2 ··· 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 ··· 12

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 D6 Dic3 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Dic3 C3×C3⋊D4 C4.D4 C12.D4 C3×C4.D4 C3×C12.D4 kernel C3×C12.D4 C3×C4.Dic3 D4×C3×C6 C12.D4 C2×C62 C4.Dic3 C6×D4 C22×C6 C6×D4 C3×C12 C2×C12 C22×C6 C2×D4 C12 C12 C2×C4 C23 C4 C32 C3 C3 C1 # reps 1 2 1 2 4 4 2 8 1 2 1 2 2 4 4 2 4 8 1 2 2 4

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 1 0 2 0 5 4 3 6 6 6 2 1 6 3 0
,
 6 4 3 5 6 6 3 0 2 5 6 6 2 2 5 3
,
 4 0 3 0 4 5 0 5 5 2 1 1 5 5 1 4
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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