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G = M4(2)⋊4C4order 64 = 26

4th semidirect product of M4(2) and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M4(2)⋊4C4, C22.3C42, (C2×C8)⋊2C4, C4.5(C4⋊C4), (C2×C4).3Q8, (C2×C4).114D4, C22⋊C4.1C4, C23.6(C2×C4), C22.6(C4⋊C4), C4.22(C22⋊C4), C42⋊C2.3C2, (C2×M4(2)).9C2, C22.8(C22⋊C4), (C22×C4).21C22, C2.9(C2.C42), (C2×C4).65(C2×C4), SmallGroup(64,25)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M4(2)⋊4C4
C1C2C4C2×C4C22×C4C2×M4(2) — M4(2)⋊4C4
C1C2C22 — M4(2)⋊4C4
C1C4C22×C4 — M4(2)⋊4C4
C1C2C2C22×C4 — M4(2)⋊4C4

Generators and relations for M4(2)⋊4C4
 G = < a,b,c | a8=b2=c4=1, bab=a5, cac-1=ab, cbc-1=a4b >

2C2
2C2
2C2
4C4
4C4
4C22
2C8
2C8
2C8
2C8
2C2×C4
2C2×C4
2M4(2)
2M4(2)
2C42
2C2×C8
2M4(2)
2C4⋊C4

Character table of M4(2)⋊4C4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 1122211222444444444444
ρ11111111111111111111111    trivial
ρ21111111111-1-1-1-11-1-1111-1-1    linear of order 2
ρ31111111111-1-1-1-1-111-1-1-111    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111-1-111-11-1-ii-ii-i11-iii-1-1    linear of order 4
ρ6111-1-1-1-11-11i-i-ii1i-i-1-11-ii    linear of order 4
ρ7111-1-111-11-1-ii-iii-1-1i-i-i11    linear of order 4
ρ8111-1-1-1-11-11i-i-ii-1-ii11-1i-i    linear of order 4
ρ9111-1-111-11-1i-ii-i-i-1-1-iii11    linear of order 4
ρ10111-1-1-1-11-11-iii-i1-ii-1-11i-i    linear of order 4
ρ1111111-1-1-1-1-111-1-1i-ii-ii-i-ii    linear of order 4
ρ1211111-1-1-1-1-1-1-111ii-i-ii-ii-i    linear of order 4
ρ13111-1-111-11-1i-ii-ii11i-i-i-1-1    linear of order 4
ρ1411111-1-1-1-1-111-1-1-ii-ii-iii-i    linear of order 4
ρ1511111-1-1-1-1-1-1-111-i-iii-ii-ii    linear of order 4
ρ16111-1-1-1-11-11-iii-i-1i-i11-1-ii    linear of order 4
ρ1722-2-22-2-2-222000000000000    orthogonal lifted from D4
ρ1822-2-22222-2-2000000000000    orthogonal lifted from D4
ρ1922-22-222-2-22000000000000    orthogonal lifted from D4
ρ2022-22-2-2-222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ214-40004i-4i000000000000000    complex faithful
ρ224-4000-4i4i000000000000000    complex faithful

Permutation representations of M4(2)⋊4C4
On 16 points - transitive group 16T90
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)
(1 9)(2 14 6 10)(3 15)(4 12 8 16)(5 13)(7 11)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,9)(2,14,6,10)(3,15)(4,12,8,16)(5,13)(7,11)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,9)(2,14,6,10)(3,15)(4,12,8,16)(5,13)(7,11) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,9),(2,14,6,10),(3,15),(4,12,8,16),(5,13),(7,11)])

G:=TransitiveGroup(16,90);

On 16 points - transitive group 16T110
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)
(1 9 5 13)(2 4)(3 15 7 11)(6 8)(10 12)(14 16)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,9,5,13)(2,4)(3,15,7,11)(6,8)(10,12)(14,16)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,9,5,13)(2,4)(3,15,7,11)(6,8)(10,12)(14,16) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10)], [(1,9,5,13),(2,4),(3,15,7,11),(6,8),(10,12),(14,16)])

G:=TransitiveGroup(16,110);

M4(2)⋊4C4 is a maximal subgroup of
C23.5C42  C8○D4⋊C4  C24.6(C2×C4)  (C2×Q8).211D4  C8⋊C417C4  M4(2).41D4  (C2×D4).Q8  M4(2).44D4  (C2×C8)⋊D4  M4(2).46D4  M4(2).47D4  C42.5D4  C42.6D4  C42.426D4  M4(2)⋊21D4  M4(2).50D4  C42.427D4  M4(2)⋊5D4  M4(2).D4  C42.8D4  M4(2).8D4  M4(2).9D4  C42.9D4  (C2×C8).D4  (C2×C8).6D4  C42.32Q8  C22⋊C4.Q8  C22⋊C4.F5  (C2×C8)⋊F5  M4(2)⋊4F5
 (C2×C4p).Q8: C8.(C4⋊C4)  M4(2).27D4  (C2×C12).Q8  (C2×C24)⋊C4  M4(2)⋊4Dic3  C20.60(C4⋊C4)  (C2×C40)⋊C4  M4(2)⋊4Dic5 ...
M4(2)⋊4C4 is a maximal quotient of
C42.2Q8  C23.21C42  C42.3Q8  C42.23D4  C42.6Q8  C42.26D4  C42.9Q8  C42.370D4  C23.C42  C23.8C42  C42.30D4  C42.31D4  C22⋊C4.F5  (C2×C8)⋊F5  M4(2)⋊4F5
 (C2×C4).D4p: C42.5Q8  C42.389D4  C42.10Q8  (C2×C12).Q8  (C2×C24)⋊C4  M4(2)⋊4Dic3  C20.60(C4⋊C4)  (C2×C40)⋊C4 ...

Matrix representation of M4(2)⋊4C4 in GL4(𝔽5) generated by

0300
0002
1000
0040
,
0002
0040
0400
3000
,
2000
0020
0300
0003
G:=sub<GL(4,GF(5))| [0,0,1,0,3,0,0,0,0,0,0,4,0,2,0,0],[0,0,0,3,0,0,4,0,0,4,0,0,2,0,0,0],[2,0,0,0,0,0,3,0,0,2,0,0,0,0,0,3] >;

M4(2)⋊4C4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4C_4
% in TeX

G:=Group("M4(2):4C4");
// GroupNames label

G:=SmallGroup(64,25);
// by ID

G=gap.SmallGroup(64,25);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^5,c*a*c^-1=a*b,c*b*c^-1=a^4*b>;
// generators/relations

Export

Subgroup lattice of M4(2)⋊4C4 in TeX
Character table of M4(2)⋊4C4 in TeX

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