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## G = C3×D4⋊6D6order 288 = 25·32

### Direct product of C3 and D4⋊6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D4⋊6D6
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C3×S3×D4 — C3×D4⋊6D6
 Lower central C3 — C6 — C3×D4⋊6D6
 Upper central C1 — C6 — C6×D4

Generators and relations for C3×D46D6
G = < a,b,c,d,e | a3=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 810 in 359 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, 2+ 1+4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C6×D4, C6×D4, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, D46D6, C3×2+ 1+4, C3×C4○D12, C3×S3×D4, C3×D42S3, C6×C3⋊D4, D4×C3×C6, C3×D46D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2+ 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, D46D6, C3×2+ 1+4, S3×C22×C6, C3×D46D6

Permutation representations of C3×D46D6
On 24 points - transitive group 24T588
Generators in S24
(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 4 15 18)(2 13 16 5)(3 6 17 14)(7 10 19 22)(8 23 20 11)(9 12 21 24)
(2 16)(4 18)(6 14)(7 19)(9 21)(11 23)
(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 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,588);

81 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I 2J 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6U 6V ··· 6AG 6AH ··· 6AO 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R order 1 2 2 ··· 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 2 ··· 2 6 6 6 6 1 1 2 2 2 2 2 6 6 6 6 1 1 2 ··· 2 4 ··· 4 6 ··· 6 2 2 2 2 4 ··· 4 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 D6 C3×S3 S3×C6 S3×C6 S3×C6 2+ 1+4 D4⋊6D6 C3×2+ 1+4 C3×D4⋊6D6 kernel C3×D4⋊6D6 C3×C4○D12 C3×S3×D4 C3×D4⋊2S3 C6×C3⋊D4 D4×C3×C6 D4⋊6D6 C4○D12 S3×D4 D4⋊2S3 C2×C3⋊D4 C6×D4 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C32 C3 C3 C1 # reps 1 2 4 4 4 1 2 4 8 8 8 2 1 1 4 2 2 2 8 4 1 2 2 4

Matrix representation of C3×D46D6 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 0 6 4 3 0 6 3 4 4 2 6 6 1 4 5
,
 6 1 6 0 6 6 2 3 6 5 3 3 0 4 3 6
,
 4 4 1 5 5 4 6 5 6 4 6 3 4 4 5 0
,
 3 3 5 0 6 0 4 6 6 1 3 2 0 0 0 1
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,4,6,0,0,4,1,6,6,2,4,4,3,6,5],[6,6,6,0,1,6,5,4,6,2,3,3,0,3,3,6],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;

C3×D46D6 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_6D_6
% in TeX

G:=Group("C3xD4:6D6");
// GroupNames label

G:=SmallGroup(288,994);
// by ID

G=gap.SmallGroup(288,994);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,555,1571,9414]);
// Polycyclic

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

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