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G = C32⋊M4(2)  order 144 = 24·32

The semidirect product of C32 and M4(2) acting via M4(2)/C4=C4

metabelian, soluble, monomial

Aliases: C32⋊1M4(2), C4.(C32⋊C4), (C3×C12).3C4, C32⋊2C8⋊3C2, C3⋊Dic3.8C22, (C2×C3⋊S3).6C4, (C4×C3⋊S3).7C2, (C3×C6).2(C2×C4), C2.4(C2×C32⋊C4), SmallGroup(144,131)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C32⋊M4(2)
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C32⋊M4(2)
Lower central
C32 — C3×C6 — C32⋊M4(2)
Upper central
C1 — C2 — C4

Generators and relations for C32⋊M4(2)
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=ab-1, dad=a-1, cbc-1=a-1b-1, dbd=b-1, dcd=c5 >

C1
C2
18C2
2C3
2C3
C4
9C4
9C22
2C6
2C6
6S3
6S3
6S3
6S3
C32
9C2×C4
9C8
9C8
2C12
2C12
6D6
6D6
6Dic3
6Dic3
C3×C6
2C3⋊S3
9M4(2)
6C4×S3
6C4×S3
C3⋊Dic3
C3×C12
C2×C3⋊S3
C32⋊2C8
C32⋊2C8
C4×C3⋊S3
C32⋊M4(2)

Character table of C32⋊M4(2)

 class 12A2B3A3B4A4B4C6A6B8A8B8C8D12A12B12C12D
 size 11184429944181818184444
ρ1111111111111111111    trivial
ρ211-111-11111-11-11-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-11111    linear of order 2
ρ411-111-111111-11-1-1-1-1-1    linear of order 2
ρ511111-1-1-111-iii-i-1-1-1-1    linear of order 4
ρ611-1111-1-111-i-iii1111    linear of order 4
ρ711-1111-1-111ii-i-i1111    linear of order 4
ρ811111-1-1-111i-i-ii-1-1-1-1    linear of order 4
ρ92-20220-2i2i-2-200000000    complex lifted from M4(2)
ρ102-202202i-2i-2-200000000    complex lifted from M4(2)
ρ11440-21-4001-20000-1-122    orthogonal lifted from C2×C32⋊C4
ρ124401-2400-210000-2-211    orthogonal lifted from C32⋊C4
ρ134401-2-400-21000022-1-1    orthogonal lifted from C2×C32⋊C4
ρ14440-214001-2000011-2-2    orthogonal lifted from C32⋊C4
ρ154-40-21000-120000-3i3i00    complex faithful
ρ164-401-20002-10000003i-3i    complex faithful
ρ174-40-21000-1200003i-3i00    complex faithful
ρ184-401-20002-1000000-3i3i    complex faithful

Permutation representations of C32⋊M4(2)
►On 24 points - transitive group 24T243
Generators in S24
(1 11 22)(2 12 23)(3 24 13)(4 17 14)(5 15 18)(6 16 19)(7 20 9)(8 21 10)

(2 23 12)(4 14 17)(6 19 16)(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)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)
 
G:=sub<Sym(24)| (1,11,22)(2,12,23)(3,24,13)(4,17,14)(5,15,18)(6,16,19)(7,20,9)(8,21,10), (2,23,12)(4,14,17)(6,19,16)(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)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23)>;
 
G:=Group( (1,11,22)(2,12,23)(3,24,13)(4,17,14)(5,15,18)(6,16,19)(7,20,9)(8,21,10), (2,23,12)(4,14,17)(6,19,16)(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)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23) );
 
G=PermutationGroup([[(1,11,22),(2,12,23),(3,24,13),(4,17,14),(5,15,18),(6,16,19),(7,20,9),(8,21,10)], [(2,23,12),(4,14,17),(6,19,16),(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),(9,20),(10,17),(11,22),(12,19),(13,24),(14,21),(15,18),(16,23)]])
 
G:=TransitiveGroup(24,243);
 

C32⋊M4(2) is a maximal subgroup of
 C4.S3≀C2  (C3×C12).D4  (C3×C24).C4  C8.(C32⋊C4)  C32⋊6C4≀C2  C32⋊7C4≀C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  C3⋊S3⋊M4(2)  C62.(C2×C4)  C12⋊S3.C4  C33⋊2M4(2)  C33⋊4M4(2)
C32⋊M4(2) is a maximal quotient of
 (C3×C12)⋊4C8  C32⋊2C8⋊C4  C62.6(C2×C4)  He3⋊1M4(2)  C33⋊2M4(2)  C33⋊4M4(2)

Matrix representation of C32⋊M4(2) ►in GL4(𝔽5) generated by

1433
2212
4141
4211
,
3310
1144
2320
0110
,
3243
2442
1042
1404
,
1324
2242
4101
4142
G:=sub<GL(4,GF(5))| [1,2,4,4,4,2,1,2,3,1,4,1,3,2,1,1],[3,1,2,0,3,1,3,1,1,4,2,1,0,4,0,0],[3,2,1,1,2,4,0,4,4,4,4,0,3,2,2,4],[1,2,4,4,3,2,1,1,2,4,0,4,4,2,1,2] >;
 

C32⋊M4(2) in GAP, Magma, Sage, TeX 

C_3^2\rtimes M_4(2)
 
% in TeX
 
G:=Group("C3^2:M4(2)");
 
// GroupNames label
 
G:=SmallGroup(144,131);
 
// by ID
 
G=gap.SmallGroup(144,131);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,55,50,3364,256,4613,881]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=b^-1,d*c*d=c^5>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊M4(2) in TeX
Character table of C32⋊M4(2) in TeX

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