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G = M5(2)⋊C2order 64 = 26

6th semidirect product of M5(2) and C2 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D8.2C4, C4.11D8, C8.16D4, M5(2)⋊6C2, C22.3SD16, C8.2(C2×C4), (C2×D8).5C2, (C2×C4).11D4, C8.C42C2, C4.5(C22⋊C4), (C2×C8).10C22, C2.10(D4⋊C4), 2-Sylow(PSigmaU(2,81)), SmallGroup(64,42)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — M5(2)⋊C2
C1C2C4C2×C4C2×C8C2×D8 — M5(2)⋊C2
C1C2C4C8 — M5(2)⋊C2
C1C2C2×C4C2×C8 — M5(2)⋊C2
C1C2C2C2C2C4C4C2×C8 — M5(2)⋊C2

Generators and relations for M5(2)⋊C2
 G = < a,b,c | a16=b2=c2=1, bab=a9, cac=a3b, bc=cb >

2C2
8C2
8C2
4C22
4C22
8C22
8C22
2D4
2D4
4D4
4C23
4C8
2C16
2M4(2)
2D8
2C2×D4

Character table of M5(2)⋊C2

 class 12A2B2C2D4A4B8A8B8C8D8E16A16B16C16D
 size 1128822224884444
ρ11111111111111111    trivial
ρ2111-1-111111-1-11111    linear of order 2
ρ3111-1-11111111-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-1-1-1    linear of order 2
ρ511-1-11-11-1-11i-i-ii-ii    linear of order 4
ρ611-11-1-11-1-11-ii-ii-ii    linear of order 4
ρ711-11-1-11-1-11i-ii-ii-i    linear of order 4
ρ811-1-11-11-1-11-iii-ii-i    linear of order 4
ρ922-200-2222-2000000    orthogonal lifted from D4
ρ102220022-2-2-2000000    orthogonal lifted from D4
ρ1122-2002-200000-222-2    orthogonal lifted from D8
ρ1222-2002-2000002-2-22    orthogonal lifted from D8
ρ1322200-2-200000-2-2--2--2    complex lifted from SD16
ρ1422200-2-200000--2--2-2-2    complex lifted from SD16
ρ154-400000-22220000000    orthogonal faithful
ρ164-40000022-220000000    orthogonal faithful

Permutation representations of M5(2)⋊C2
On 16 points - transitive group 16T144
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 9)(3 11)(5 13)(7 15)
(2 4)(3 15)(5 13)(6 16)(7 11)(8 14)(10 12)

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

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

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

G:=TransitiveGroup(16,144);

M5(2)⋊C2 is a maximal subgroup of
C23.21SD16  D16⋊C4  Q16.10D4  D8.3D4  D4.3D8  D4.5D8  M5(2).C22  D40.C4
 C4p.D8: C8.3D8  D83D4  D24.C4  M5(2)⋊S3  D8.Dic3  D40.6C4  D40.4C4  D8.Dic5 ...
M5(2)⋊C2 is a maximal quotient of
D8⋊C8  C23.12SD16  C4.6Q32  C8.2C42  D40.C4
 C4p.D8: C4.D16  D24.C4  M5(2)⋊S3  D8.Dic3  D40.6C4  D40.4C4  D8.Dic5  D56.C4 ...

Matrix representation of M5(2)⋊C2 in GL4(𝔽7) generated by

2340
0640
2152
4501
,
1063
0232
0322
0112
,
1610
0365
0455
0235
G:=sub<GL(4,GF(7))| [2,0,2,4,3,6,1,5,4,4,5,0,0,0,2,1],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2],[1,0,0,0,6,3,4,2,1,6,5,3,0,5,5,5] >;

M5(2)⋊C2 in GAP, Magma, Sage, TeX

M_5(2)\rtimes C_2
% in TeX

G:=Group("M5(2):C2");
// GroupNames label

G:=SmallGroup(64,42);
// by ID

G=gap.SmallGroup(64,42);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,476,86,489,117,1444,730,88]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^2=1,b*a*b=a^9,c*a*c=a^3*b,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of M5(2)⋊C2 in TeX
Character table of M5(2)⋊C2 in TeX

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