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## G = M5(2).C22order 128 = 27

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

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

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, C82D4, 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 1 2A 2B 2C 2D 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 16A 16B 16C 16D size 1 1 2 4 16 2 2 4 16 4 4 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 -1 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 1 1 linear of order 2 ρ7 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 0 2 2 2 0 -2 -2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 2 2 -2 0 -2 -2 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 -2 2 0 0 -2 2 -2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ12 2 2 -2 0 0 -2 2 0 0 2 -2 0 2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ13 2 2 -2 0 0 -2 2 0 0 -2 2 2i 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 -2 0 0 -2 2 0 0 2 -2 0 -2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ15 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ16 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ17 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ18 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ19 4 4 -4 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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)

 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 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 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

`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`

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