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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: C321M4(2), C4.(C32⋊C4), (C3×C12).3C4, C322C83C2, 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

C1C3×C6 — C32⋊M4(2)
C1C32C3×C6C3⋊Dic3C322C8 — C32⋊M4(2)
C32C3×C6 — C32⋊M4(2)
C1C2C4

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 >

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

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

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

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

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

G:=TransitiveGroup(24,243);

C32⋊M4(2) is a maximal subgroup of
C4.S3≀C2  (C3×C12).D4  (C3×C24).C4  C8.(C32⋊C4)  C326C4≀C2  C327C4≀C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  C3⋊S3⋊M4(2)  C62.(C2×C4)  C12⋊S3.C4  C332M4(2)  C334M4(2)
C32⋊M4(2) is a maximal quotient of
(C3×C12)⋊4C8  C322C8⋊C4  C62.6(C2×C4)  He31M4(2)  C332M4(2)  C334M4(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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