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G = C3312M4(2)  order 432 = 24·33

2nd semidirect product of C33 and M4(2) acting via M4(2)/C22=C4

metabelian, soluble, monomial

Aliases: C3312M4(2), C62.13Dic3, C334C89C2, (C3×C62).4C4, C3⋊Dic3.43D6, C3⋊Dic3.8Dic3, C22.(C33⋊C4), C32(C62.C4), C327(C4.Dic3), C6.13(C2×C32⋊C4), (C2×C6).5(C32⋊C4), (C3×C3⋊Dic3).7C4, C2.6(C2×C33⋊C4), (C2×C3⋊Dic3).11S3, (C6×C3⋊Dic3).11C2, (C32×C6).20(C2×C4), (C3×C6).27(C2×Dic3), (C3×C3⋊Dic3).51C22, SmallGroup(432,640)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C3312M4(2)
C1C3C33C32×C6C3×C3⋊Dic3C334C8 — C3312M4(2)
C33C32×C6 — C3312M4(2)
C1C2C22

Generators and relations for C3312M4(2)
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=ab-1, ae=ea, bc=cb, dbd-1=a-1b-1, be=eb, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 392 in 92 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, C6, C6 [×13], C8 [×2], C2×C4, C32, C32 [×4], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], M4(2), C3×C6, C3×C6 [×13], C3⋊C8 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C62 [×4], C4.Dic3, C32×C6, C32×C6, C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C3×C62, C62.C4, C334C8 [×2], C6×C3⋊Dic3, C3312M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, M4(2), C2×Dic3, C32⋊C4, C4.Dic3, C2×C32⋊C4, C33⋊C4, C62.C4, C2×C33⋊C4, C3312M4(2)

Permutation representations of C3312M4(2)
On 24 points - transitive group 24T1287
Generators in S24
(1 22 11)(2 12 23)(3 13 24)(4 17 14)(5 18 15)(6 16 19)(7 9 20)(8 21 10)
(1 11 22)(3 24 13)(5 15 18)(7 20 9)
(1 22 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 21)
(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)| (1,22,11)(2,12,23)(3,13,24)(4,17,14)(5,18,15)(6,16,19)(7,9,20)(8,21,10), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (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( (1,22,11)(2,12,23)(3,13,24)(4,17,14)(5,18,15)(6,16,19)(7,9,20)(8,21,10), (1,11,22)(3,24,13)(5,15,18)(7,20,9), (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (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([(1,22,11),(2,12,23),(3,13,24),(4,17,14),(5,18,15),(6,16,19),(7,9,20),(8,21,10)], [(1,11,22),(3,24,13),(5,15,18),(7,20,9)], [(1,22,11),(2,12,23),(3,24,13),(4,14,17),(5,18,15),(6,16,19),(7,20,9),(8,10,21)], [(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(24,1287);

42 conjugacy classes

class 1 2A2B3A3B···3G4A4B4C6A6B6C6D···6U8A8B8C8D12A12B12C12D
order12233···34446666···6888812121212
size11224···499182224···45454545418181818

42 irreducible representations

dim11111222222444444
type++++-+-++-
imageC1C2C2C4C4S3Dic3D6Dic3M4(2)C4.Dic3C32⋊C4C2×C32⋊C4C33⋊C4C62.C4C2×C33⋊C4C3312M4(2)
kernelC3312M4(2)C334C8C6×C3⋊Dic3C3×C3⋊Dic3C3×C62C2×C3⋊Dic3C3⋊Dic3C3⋊Dic3C62C33C32C2×C6C6C22C3C2C1
# reps12122111124224448

Matrix representation of C3312M4(2) in GL4(𝔽7) generated by

3243
4556
3361
0001
,
0526
0202
3361
0004
,
3632
6342
0020
0004
,
3610
3314
4355
2253
,
0632
6042
0060
0001
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[3,3,4,2,6,3,3,2,1,1,5,5,0,4,5,3],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1] >;

C3312M4(2) in GAP, Magma, Sage, TeX

C_3^3\rtimes_{12}M_4(2)
% in TeX

G:=Group("C3^3:12M4(2)");
// GroupNames label

G:=SmallGroup(432,640);
// by ID

G=gap.SmallGroup(432,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,58,2804,298,2693,1027,14118]);
// Polycyclic

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

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