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G = C33⋊2M4(2)  order 432 = 24·33

2nd semidirect product of C33 and M4(2) acting via M4(2)/C2=C2×C4

metabelian, soluble, monomial

Aliases: C33⋊2M4(2), C33⋊4C8⋊7C2, C32⋊2C8⋊4S3, C3⋊Dic3.26D6, Dic3.(C32⋊C4), C32⋊5(C8⋊S3), (C32×Dic3).2C4, C3⋊1(C32⋊M4(2)), C2.7(S3×C32⋊C4), C6.7(C2×C32⋊C4), (C3×C6).32(C4×S3), C33⋊8(C2×C4).4C2, (C3×C32⋊2C8)⋊7C2, (C32×C6).7(C2×C4), (C2×C33⋊C2).2C4, (C3×C3⋊Dic3).29C22, SmallGroup(432,573)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊2M4(2)
Chief series
C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8(C2×C4) — C33⋊2M4(2)
Lower central
C33 — C32×C6 — C33⋊2M4(2)
Upper central
C1 — C2

Generators and relations for C33⋊2M4(2)
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, eae=a-1, bc=cb, ebe=b-1, dcd-1=ece=c-1, ede=d5 >

Subgroups: 864 in 92 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, Dic3, C12, D6, M4(2), C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, C33⋊C2, C32×C6, C32⋊2C8, C32⋊2C8, C6.D6, C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C32⋊M4(2), C3×C32⋊2C8, C33⋊4C8, C33⋊8(C2×C4), C33⋊2M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C4×S3, C32⋊C4, C8⋊S3, C2×C32⋊C4, C32⋊M4(2), S3×C32⋊C4, C33⋊2M4(2)

Character table of C33⋊2M4(2)

 class 12A2B3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A24B24C24D
 size 115424488699244881818545412121212181818181818
ρ1111111111111111111111111111111    trivial
ρ211-111111-1111111111-1-1-1-1-1-1111111    linear of order 2
ρ31111111111111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ411-111111-11111111-1-111-1-1-1-111-1-1-1-1    linear of order 2
ρ511-1111111-1-111111i-i-ii1111-1-1-iii-i    linear of order 4
ρ611111111-1-1-111111i-ii-i-1-1-1-1-1-1-iii-i    linear of order 4
ρ711-1111111-1-111111-iii-i1111-1-1i-i-ii    linear of order 4
ρ811111111-1-1-111111-ii-ii-1-1-1-1-1-1i-i-ii    linear of order 4
ρ9220-122-1-1022-122-1-1-2-2000000-1-11111    orthogonal lifted from D6
ρ10220-122-1-1022-122-1-122000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ11220-122-1-10-2-2-122-1-12i-2i00000011i-i-ii    complex lifted from C4×S3
ρ122-20222220-2i2i-2-2-2-2-2000000002i-2i0000    complex lifted from M4(2)
ρ132-202222202i-2i-2-2-2-2-200000000-2i2i0000    complex lifted from M4(2)
ρ14220-122-1-10-2-2-122-1-1-2i2i00000011-iii-i    complex lifted from C4×S3
ρ152-20-122-1-102i-2i1-2-21100000000i-i2ζ83ζ3+ζ832ζ85ζ3+ζ852ζ8ζ3+ζ82ζ87ζ3+ζ87    complex lifted from C8⋊S3
ρ162-20-122-1-10-2i2i1-2-21100000000-ii2ζ85ζ3+ζ852ζ83ζ3+ζ832ζ87ζ3+ζ872ζ8ζ3+ζ8    complex lifted from C8⋊S3
ρ172-20-122-1-10-2i2i1-2-21100000000-ii2ζ8ζ3+ζ82ζ87ζ3+ζ872ζ83ζ3+ζ832ζ85ζ3+ζ85    complex lifted from C8⋊S3
ρ182-20-122-1-102i-2i1-2-21100000000i-i2ζ87ζ3+ζ872ζ8ζ3+ζ82ζ85ζ3+ζ852ζ83ζ3+ζ83    complex lifted from C8⋊S3
ρ1944041-2-214004-21-210000-2-211000000    orthogonal lifted from C32⋊C4
ρ204404-211-240041-21-2000011-2-2000000    orthogonal lifted from C32⋊C4
ρ2144041-2-21-4004-21-21000022-1-1000000    orthogonal lifted from C2×C32⋊C4
ρ224404-211-2-40041-21-20000-1-122000000    orthogonal lifted from C2×C32⋊C4
ρ234-4041-2-21000-42-12-1000000-3i3i000000    complex lifted from C32⋊M4(2)
ρ244-404-211-2000-4-12-1200003i-3i00000000    complex lifted from C32⋊M4(2)
ρ254-404-211-2000-4-12-120000-3i3i00000000    complex lifted from C32⋊M4(2)
ρ264-4041-2-21000-42-12-10000003i-3i000000    complex lifted from C32⋊M4(2)
ρ27880-42-42-1000-4-422-100000000000000    orthogonal lifted from S3×C32⋊C4
ρ28880-4-42-12000-42-4-1200000000000000    orthogonal lifted from S3×C32⋊C4
ρ298-80-42-42-100044-2-2100000000000000    orthogonal faithful
ρ308-80-4-42-120004-241-200000000000000    orthogonal faithful

Permutation representations of C33⋊2M4(2)
►On 24 points - transitive group 24T1309
Generators in S24
(1 13 22)(2 14 23)(3 24 15)(4 17 16)(5 9 18)(6 10 19)(7 20 11)(8 21 12)

(1 22 13)(3 15 24)(5 18 9)(7 11 20)

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

(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)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
 
G:=sub<Sym(24)| (1,13,22)(2,14,23)(3,24,15)(4,17,16)(5,9,18)(6,10,19)(7,20,11)(8,21,12), (1,22,13)(3,15,24)(5,18,9)(7,11,20), (1,13,22)(2,23,14)(3,15,24)(4,17,16)(5,9,18)(6,19,10)(7,11,20)(8,21,12), (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)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;
 
G:=Group( (1,13,22)(2,14,23)(3,24,15)(4,17,16)(5,9,18)(6,10,19)(7,20,11)(8,21,12), (1,22,13)(3,15,24)(5,18,9)(7,11,20), (1,13,22)(2,23,14)(3,15,24)(4,17,16)(5,9,18)(6,19,10)(7,11,20)(8,21,12), (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)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );
 
G=PermutationGroup([[(1,13,22),(2,14,23),(3,24,15),(4,17,16),(5,9,18),(6,10,19),(7,20,11),(8,21,12)], [(1,22,13),(3,15,24),(5,18,9),(7,11,20)], [(1,13,22),(2,23,14),(3,15,24),(4,17,16),(5,9,18),(6,19,10),(7,11,20),(8,21,12)], [(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),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)]])
 
G:=TransitiveGroup(24,1309);
 

Matrix representation of C33⋊2M4(2) ►in GL6(𝔽73)

100000
010000
0072010
0000072
0072000
0001072
,
100000
010000
00727212
0071012
0072101
00727122
,
0720000
1720000
001000
000100
000010
000001
,
6700000
3670000
0000072
0072000
0071011
00172072
,
0720000
7200000
0072101
000001
00727212
000100

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,71,72,72,0,0,72,0,1,71,0,0,1,1,0,2,0,0,2,2,1,2],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,3,0,0,0,0,70,67,0,0,0,0,0,0,0,72,71,1,0,0,0,0,0,72,0,0,0,0,1,0,0,0,72,0,1,72],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,72,0,0,0,1,0,72,1,0,0,0,0,1,0,0,0,1,1,2,0] >;
 

C33⋊2M4(2) in GAP, Magma, Sage, TeX 

C_3^3\rtimes_2M_4(2)
 
% in TeX
 
G:=Group("C3^3:2M4(2)");
 
// GroupNames label
 
G:=SmallGroup(432,573);
 
// by ID
 
G=gap.SmallGroup(432,573);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,36,58,1411,298,1356,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,e*a*e=a^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
 
// generators/relations
 

Export

Character table of C33⋊2M4(2) in TeX

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