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## G = C3×C24⋊4S3order 288 = 25·32

### Direct product of C3 and C24⋊4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C24⋊4S3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×C3⋊D4 — C3×C24⋊4S3
 Lower central C3 — C2×C6 — C3×C24⋊4S3
 Upper central C1 — C2×C6 — C23×C6

Generators and relations for C3×C244S3
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 714 in 327 conjugacy classes, 82 normal (16 characteristic)
C1, C2 [×3], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×6], C22 [×17], S3, C6 [×6], C6 [×28], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], C32, Dic3 [×3], C12 [×3], D6 [×3], C2×C6 [×2], C2×C6 [×12], C2×C6 [×66], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3×C6 [×3], C3×C6 [×6], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×6], C22×S3, C22×C6 [×6], C22×C6 [×28], C22≀C2, C3×Dic3 [×3], S3×C6 [×3], C62, C62 [×6], C62 [×14], C6.D4 [×3], C3×C22⋊C4 [×3], C2×C3⋊D4 [×3], C6×D4 [×3], C23×C6 [×2], C23×C6, C6×Dic3 [×3], C3×C3⋊D4 [×6], S3×C2×C6, C2×C62 [×3], C2×C62 [×6], C244S3, C3×C22≀C2, C3×C6.D4 [×3], C6×C3⋊D4 [×3], C22×C62, C3×C244S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×6], C23, D6 [×3], C2×C6 [×7], C2×D4 [×3], C3×S3, C3⋊D4 [×6], C3×D4 [×6], C22×S3, C22×C6, C22≀C2, S3×C6 [×3], C2×C3⋊D4 [×3], C6×D4 [×3], C3×C3⋊D4 [×6], S3×C2×C6, C244S3, C3×C22≀C2, C6×C3⋊D4 [×3], C3×C244S3

Permutation representations of C3×C244S3
On 24 points - transitive group 24T626
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)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)

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

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

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

G:=TransitiveGroup(24,626);

90 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 3A 3B 3C 3D 3E 4A 4B 4C 6A ··· 6F 6G ··· 6BK 6BL 6BM 12A ··· 12F order 1 2 2 2 2 ··· 2 2 3 3 3 3 3 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 1 1 2 ··· 2 12 1 1 2 2 2 12 12 12 1 ··· 1 2 ··· 2 12 12 12 ··· 12

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C3×C24⋊4S3 C3×C6.D4 C6×C3⋊D4 C22×C62 C24⋊4S3 C6.D4 C2×C3⋊D4 C23×C6 C23×C6 C62 C22×C6 C24 C2×C6 C2×C6 C23 C22 # reps 1 3 3 1 2 6 6 2 1 6 3 2 12 12 6 24

Matrix representation of C3×C244S3 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 9
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C244S3 in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes_4S_3
% in TeX

G:=Group("C3xC2^4:4S3");
// GroupNames label

G:=SmallGroup(288,724);
// by ID

G=gap.SmallGroup(288,724);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,9414]);
// Polycyclic

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

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