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G = M5(2).C22  order 128 = 27

8th non-split extension by M5(2) of C22 acting faithfully

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

Aliases: C23.9SD16, M5(2).8C22, C8.Q8⋊5C2, (C2×C8).13D4, C8⋊2D4.3C2, C23.C8⋊3C2, (C2×C8).5C23, (C2×C4).9SD16, M5(2)⋊C2⋊5C2, C8.27(C4○D4), C4.99(C8⋊C22), C4.Q8.5C22, (C2×D8).51C22, (C22×C4).106D4, C8.C4.8C22, M4(2).C4⋊10C2, C22.21(C2×SD16), C4.45(C22.D4), (C2×M4(2)).37C22, C2.12(C23.46D4), (C2×C4).285(C2×D4), SmallGroup(128,970)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — M5(2).C22
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C8.C4 — M4(2).C4 — M5(2).C22
Lower central
C1 — C2 — C4 — C2×C8 — M5(2).C22
Upper central
C1 — C2 — C2×C4 — C2×M4(2) — M5(2).C22
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — M5(2).C22

Generators and relations for M5(2).C22
 G = < a,b,c,d | a16=b2=c2=1, d2=b, bab=a9, cac=ab, dad-1=a11, dcd-1=bc=cb, bd=db >

Subgroups: 180 in 67 conjugacy classes, 28 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, D4⋊C4, C4.Q8, C8.C4, C8.C4, M5(2), C4⋊D4, C2×M4(2), C2×M4(2), C2×D8, C23.C8, M5(2)⋊C2, C8.Q8, M4(2).C4, C8⋊2D4, M5(2).C22
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C22.D4, C2×SD16, C8⋊C22, C23.46D4, M5(2).C22

Character table of M5(2).C22

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G16A16B16C16D
 size 1124162241644888888888
ρ111111111111111111111    trivial
ρ2111-1-111-1111-11-1-111-11-1    linear of order 2
ρ31111-1111-11111111-1-1-1-1    linear of order 2
ρ4111-1111-1-111-11-1-11-11-11    linear of order 2
ρ5111-1111-1-1111-11-1-11-11-1    linear of order 2
ρ61111-1111-111-1-1-11-11111    linear of order 2
ρ7111-1-111-11111-11-1-1-11-11    linear of order 2
ρ811111111111-1-1-11-1-1-1-1-1    linear of order 2
ρ9222202220-2-2000-200000    orthogonal lifted from D4
ρ10222-2022-20-2-2000200000    orthogonal lifted from D4
ρ1122-200-2200-22-2i02i000000    complex lifted from C4○D4
ρ1222-200-22002-202i00-2i0000    complex lifted from C4○D4
ρ1322-200-2200-222i0-2i000000    complex lifted from C4○D4
ρ1422-200-22002-20-2i002i0000    complex lifted from C4○D4
ρ15222-20-2-2200000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ1622220-2-2-200000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ17222-20-2-2200000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ1822220-2-2-200000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1944-4004-40000000000000    orthogonal lifted from C8⋊C22
ρ208-8000000000000000000    orthogonal faithful

Permutation representations of M5(2).C22
►On 16 points - transitive group 16T374
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(2 10)(4 12)(6 14)(8 16)

(1 9)(4 12)(5 13)(8 16)

(1 9)(2 12 10 4)(3 15)(6 8 14 16)(7 11)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16), (1,9)(4,12)(5,13)(8,16), (1,9)(2,12,10,4)(3,15)(6,8,14,16)(7,11)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16), (1,9)(4,12)(5,13)(8,16), (1,9)(2,12,10,4)(3,15)(6,8,14,16)(7,11) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,10),(4,12),(6,14),(8,16)], [(1,9),(4,12),(5,13),(8,16)], [(1,9),(2,12,10,4),(3,15),(6,8,14,16),(7,11)]])
 
G:=TransitiveGroup(16,374);
 

Matrix representation of M5(2).C22 ►in GL8(ℤ)

00000010
0000000-1
0000-1000
00000100
01000000
10000000
00100000
000-10000
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00-100000
000-10000
0000-1000
00000-100
00000010
00000001
,
10000000
0-1000000
000-10000
00-100000
000000-10
00000001
00001000
00000-100

G:=sub<GL(8,Integers())| [0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0] >;
 

M5(2).C22 in GAP, Magma, Sage, TeX 

M_5(2).C_2^2
 
% in TeX
 
G:=Group("M5(2).C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,970);
 
// by ID
 
G=gap.SmallGroup(128,970);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,226,521,1684,1411,998,242,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^16=b^2=c^2=1,d^2=b,b*a*b=a^9,c*a*c=a*b,d*a*d^-1=a^11,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

Export

Character table of M5(2).C22 in TeX

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