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G = M4(2).C4order 64 = 26

1st non-split extension by M4(2) of C4 acting via C4/C2=C2

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

Aliases: C23.5Q8, M4(2).1C4, M4(2).10C22, C8.4(C2×C4), (C2×C4).7Q8, C4.75(C2×D4), C4.17(C4⋊C4), C8.C43C2, (C2×C4).129D4, C22.2(C2×Q8), (C2×C8).14C22, (C2×C4).72C23, C4.29(C22×C4), C22.11(C4⋊C4), (C2×M4(2)).2C2, (C22×C4).39C22, C2.16(C2×C4⋊C4), (C2×C4).27(C2×C4), SmallGroup(64,111)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2).C4
C1C2C4C2×C4C22×C4C2×M4(2) — M4(2).C4
C1C2C4 — M4(2).C4
C1C4C22×C4 — M4(2).C4
C1C2C2C2×C4 — M4(2).C4

Generators and relations for M4(2).C4
 G = < a,b,c | a8=b2=1, c4=a4, bab=a5, cac-1=a-1, cbc-1=a4b >

2C2
2C2
2C2
4C22
2C8
2C8
2C8
2C8
2C2×C8
2C2×C8
2M4(2)
2M4(2)

Character table of M4(2).C4

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J8K8L
 size 1122211222444444444444
ρ11111111111111111111111    trivial
ρ2111-1-111-11-11-111-111-1-11-1-1    linear of order 2
ρ311111111111-1-1-1-1-1-1-111-11    linear of order 2
ρ4111-1-111-11-111-1-11-1-11-111-1    linear of order 2
ρ5111-1-111-11-1-1-11-1-11-111-111    linear of order 2
ρ61111111111-111-111-1-1-1-1-1-1    linear of order 2
ρ7111-1-111-11-1-11-111-11-11-1-11    linear of order 2
ρ81111111111-1-1-11-1-111-1-11-1    linear of order 2
ρ911-1-11-1-111-1-i-1-1i11-i-iiii-i    linear of order 4
ρ1011-11-1-1-1-111i1-1-i-11i-ii-ii-i    linear of order 4
ρ1111-1-11-1-111-1i-1-1-i11ii-i-i-ii    linear of order 4
ρ1211-11-1-1-1-111-i1-1i-11-ii-ii-ii    linear of order 4
ρ1311-11-1-1-1-111i-11i1-1-iii-i-i-i    linear of order 4
ρ1411-1-11-1-111-1-i11-i-1-1iiii-i-i    linear of order 4
ρ1511-11-1-1-1-111-i-11-i1-1i-i-iiii    linear of order 4
ρ1611-1-11-1-111-1i11i-1-1-i-i-i-iii    linear of order 4
ρ1722-2-2222-2-22000000000000    orthogonal lifted from D4
ρ1822-22-2222-2-2000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ20222-2-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ214-40004i-4i000000000000000    complex faithful
ρ224-4000-4i4i000000000000000    complex faithful

Permutation representations of M4(2).C4
On 16 points - transitive group 16T103
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(3 7)(10 14)(12 16)
(1 9 7 11 5 13 3 15)(2 16 8 10 6 12 4 14)

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

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

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

G:=TransitiveGroup(16,103);

M4(2).C4 is a maximal subgroup of
M4(2).29C23  M4(2).1F5
 M4(2).D2p: M4(2).40D4  M4(2).41D4  C42.427D4  M4(2).27D4  M4(2).8D4  M4(2).9D4  C24.Q8  M4(2).15D4 ...
 (C2×C8).D2p: (C2×C8).D4  (C2×C8).6D4  C24.11Q8  C42.10D4  C42.32Q8  C22⋊C4.Q8  M5(2).C22  C23.10SD16 ...
M4(2).C4 is a maximal quotient of
M4(2)⋊1C8  C81M4(2)  C42.90D4  C42.91D4  C42.Q8  C24.7Q8  C8.6C42  C42.104D4  M4(2).1F5
 M4(2).D2p: C24.10Q8  C42.430D4  M4(2).25D6  C23.8Dic6  M4(2).25D10  C23.Dic10  M4(2).25D14  C23.Dic14 ...
 (C2×C8).D2p: C24.9Q8  C42.106D4  C23.9Dic6  M4(2).Dic5  M4(2).Dic7 ...

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

0003
0010
0200
1000
,
1000
0400
0010
0004
,
0300
4000
0004
0030
G:=sub<GL(4,GF(5))| [0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,4,0,0,3,0,0,0,0,0,0,3,0,0,4,0] >;

M4(2).C4 in GAP, Magma, Sage, TeX

M_4(2).C_4
% in TeX

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

G:=SmallGroup(64,111);
// by ID

G=gap.SmallGroup(64,111);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,963,117,88]);
// Polycyclic

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

Export

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

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