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G = C3×C244S3order 288 = 25·32

Direct product of C3 and C244S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×C244S3, (C2×C6)C2, C6223D4, C62.211C23, (C23×C6)⋊7S3, C6.63(C6×D4), (C23×C6)⋊11C6, C2410(C3×S3), C3212C22≀C2, C23.35(S3×C6), (C22×C62)⋊3C2, C6.D413C6, (C22×C6).131D6, (C6×Dic3)⋊19C22, (C2×C62).103C22, (C2×C6)⋊11(C3×D4), (C2×C3⋊D4)⋊8C6, (S3×C2×C6)⋊5C22, C33(C3×C22≀C2), (C6×C3⋊D4)⋊22C2, C2.26(C6×C3⋊D4), C225(C3×C3⋊D4), C22.66(S3×C2×C6), (C2×C6)⋊16(C3⋊D4), (C22×S3)⋊2(C2×C6), (C2×Dic3)⋊3(C2×C6), (C3×C6).271(C2×D4), C6.164(C2×C3⋊D4), (C22×C6).67(C2×C6), (C2×C6).66(C22×C6), (C3×C6.D4)⋊29C2, (C2×C6).344(C22×S3), SmallGroup(288,724)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C244S3
C1C3C6C2×C6C62S3×C2×C6C6×C3⋊D4 — C3×C244S3
C3C2×C6 — C3×C244S3
C1C2×C6C23×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 2A2B2C2D···2I2J3A3B3C3D3E4A4B4C6A···6F6G···6BK6BL6BM12A···12F
order12222···22333334446···66···66612···12
size11112···212112221212121···12···2121212···12

90 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4
kernelC3×C244S3C3×C6.D4C6×C3⋊D4C22×C62C244S3C6.D4C2×C3⋊D4C23×C6C23×C6C62C22×C6C24C2×C6C2×C6C23C22
# reps1331266216321212624

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

1000
0100
0030
0003
,
12000
01200
0010
00012
,
1000
01200
0010
00012
,
12000
01200
00120
00012
,
1000
0100
00120
00012
,
1000
0100
0030
0009
,
0100
1000
0001
0010
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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