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G = C3⋊S3⋊M4(2)  order 288 = 25·32

2nd semidirect product of C3⋊S3 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, soluble, monomial

Aliases: (C6×C12).11C4, C3⋊S33M4(2), C62.15(C2×C4), C324(C2×M4(2)), C62.C46C2, C322C88C22, C32⋊M4(2)⋊8C2, C3⋊Dic3.31C23, C3⋊S33C88C2, (C4×C3⋊S3).17C4, C4.13(C2×C32⋊C4), (C3×C12).20(C2×C4), (C2×C4).8(C32⋊C4), C2.5(C22×C32⋊C4), C22.6(C2×C32⋊C4), (C4×C3⋊S3).97C22, (C22×C3⋊S3).16C4, C3⋊Dic3.52(C2×C4), (C3×C6).26(C22×C4), (C2×C3⋊Dic3).174C22, (C2×C4×C3⋊S3).27C2, (C2×C3⋊S3).46(C2×C4), SmallGroup(288,931)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3⋊S3⋊M4(2)
C1C32C3×C6C3⋊Dic3C322C8C3⋊S33C8 — C3⋊S3⋊M4(2)
C32C3×C6 — C3⋊S3⋊M4(2)
C1C4C2×C4

Generators and relations for C3⋊S3⋊M4(2)
 G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, cac=a-1, dad-1=ab-1, ae=ea, cbc=b-1, dbd-1=a-1b-1, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 576 in 122 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×2], C322C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S33C8 [×2], C32⋊M4(2) [×2], C62.C4 [×2], C2×C4×C3⋊S3, C3⋊S3⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, C3⋊S3⋊M4(2)

Permutation representations of C3⋊S3⋊M4(2)
On 24 points - transitive group 24T624
Generators in S24
(1 22 13)(3 15 24)(5 18 9)(7 11 20)
(1 22 13)(2 23 14)(3 15 24)(4 16 17)(5 18 9)(6 19 10)(7 11 20)(8 12 21)
(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,624);

36 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F8A···8H12A···12H
order122222334444446···68···812···12
size11299184411299184···418···184···4

36 irreducible representations

dim1111111124444
type++++++++
imageC1C2C2C2C2C4C4C4M4(2)C32⋊C4C2×C32⋊C4C2×C32⋊C4C3⋊S3⋊M4(2)
kernelC3⋊S3⋊M4(2)C3⋊S33C8C32⋊M4(2)C62.C4C2×C4×C3⋊S3C4×C3⋊S3C6×C12C22×C3⋊S3C3⋊S3C2×C4C4C22C1
# reps1222142242428

Matrix representation of C3⋊S3⋊M4(2) in GL4(𝔽5) generated by

0003
0100
0010
3004
,
0003
0410
0400
3004
,
4002
0410
0010
0001
,
0200
4002
0002
0010
,
1000
0400
0040
0001
G:=sub<GL(4,GF(5))| [0,0,0,3,0,1,0,0,0,0,1,0,3,0,0,4],[0,0,0,3,0,4,4,0,0,1,0,0,3,0,0,4],[4,0,0,0,0,4,0,0,0,1,1,0,2,0,0,1],[0,4,0,0,2,0,0,0,0,0,0,1,0,2,2,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1] >;

C3⋊S3⋊M4(2) in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes M_4(2)
% in TeX

G:=Group("C3:S3:M4(2)");
// GroupNames label

G:=SmallGroup(288,931);
// by ID

G=gap.SmallGroup(288,931);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,422,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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