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## G = C3×C23⋊A4order 288 = 25·32

### Direct product of C3 and C23⋊A4

Aliases: C3×C23⋊A4, 2+ 1+43C32, Q83(C3×A4), (C3×Q8)⋊2A4, C232(C3×A4), (C22×C6)⋊2A4, C6.5(C22⋊A4), (C3×2+ 1+4)⋊2C3, C2.2(C3×C22⋊A4), SmallGroup(288,987)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — C3×C23⋊A4
 Chief series C1 — C2 — C23 — 2+ 1+4 — C23⋊A4 — C3×C23⋊A4
 Lower central 2+ 1+4 — C3×C23⋊A4
 Upper central C1 — C6

Generators and relations for C3×C23⋊A4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, gbg-1=bc=cb, fbf=bd=db, be=eb, ece=cd=dc, cf=fc, gcg-1=b, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 480 in 114 conjugacy classes, 20 normal (7 characteristic)
C1, C2, C2 [×3], C3, C3 [×3], C4 [×2], C22 [×7], C6, C6 [×6], C2×C4 [×3], D4 [×6], Q8 [×2], C23 [×3], C23, C32, C12 [×2], A4 [×9], C2×C6 [×7], C2×D4 [×3], C4○D4 [×2], C3×C6, SL2(𝔽3) [×6], C2×C12 [×3], C3×D4 [×6], C3×Q8 [×2], C2×A4 [×9], C22×C6 [×3], C22×C6, 2+ 1+4, C3×A4 [×3], C6×D4 [×3], C3×C4○D4 [×2], C3×SL2(𝔽3) [×2], C6×A4 [×3], C23⋊A4 [×3], C3×2+ 1+4, C3×C23⋊A4
Quotients: C1, C3 [×4], C32, A4 [×5], C3×A4 [×5], C22⋊A4, C23⋊A4, C3×C22⋊A4, C3×C23⋊A4

Permutation representations of C3×C23⋊A4
On 24 points - transitive group 24T591
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)
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(4 8)(5 9)(6 7)(16 19)(17 20)(18 21)
(4 14 16)(5 15 17)(6 13 18)(7 11 21)(8 12 19)(9 10 20)

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

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

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

G:=TransitiveGroup(24,591);

33 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 6I ··· 6N 12A 12B 12C 12D order 1 2 2 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 6 6 6 1 1 16 ··· 16 6 6 1 1 6 ··· 6 16 ··· 16 6 6 6 6

33 irreducible representations

 dim 1 1 1 3 3 3 3 4 4 4 type + + + + image C1 C3 C3 A4 A4 C3×A4 C3×A4 C23⋊A4 C23⋊A4 C3×C23⋊A4 kernel C3×C23⋊A4 C23⋊A4 C3×2+ 1+4 C3×Q8 C22×C6 Q8 C23 C3 C3 C1 # reps 1 6 2 2 3 4 6 1 2 6

Matrix representation of C3×C23⋊A4 in GL4(𝔽7) generated by

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

C3×C23⋊A4 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes A_4
% in TeX

G:=Group("C3xC2^3:A4");
// GroupNames label

G:=SmallGroup(288,987);
// by ID

G=gap.SmallGroup(288,987);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,380,759,2524,375,4541,1027]);
// Polycyclic

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

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