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## G = M5(2).19C22order 128 = 27

### 6th non-split extension by M5(2) of C22 acting via C22/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — M5(2).19C22
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C8○D4 — M5(2).19C22
 Lower central C1 — C2 — C22 — M5(2).19C22
 Upper central C1 — C8 — C22×C8 — M5(2).19C22
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — M5(2).19C22

Generators and relations for M5(2).19C22
G = < a,b,c,d | a16=b2=c2=d2=1, bab=a9, cac=ab, ad=da, bc=cb, bd=db, dcd=a8c >

Subgroups: 172 in 110 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2×C16, M5(2), M5(2), C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C23.C8, C2×M5(2), C2×C8○D4, M5(2).19C22
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, M5(2).19C22

Smallest permutation representation of M5(2).19C22
On 32 points
Generators in S32
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
(2 10)(3 11)(6 14)(7 15)(17 25)(20 28)(21 29)(24 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)```

`G:=sub<Sym(32)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (2,10)(3,11)(6,14)(7,15)(17,25)(20,28)(21,29)(24,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (2,10)(3,11)(6,14)(7,15)(17,25)(20,28)(21,29)(24,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)], [(2,10),(3,11),(6,14),(7,15),(17,25),(20,28),(21,29),(24,32)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)]])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 16A ··· 16P order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C8 D4 M4(2) M5(2).19C22 kernel M5(2).19C22 C23.C8 C2×M5(2) C2×C8○D4 C22×C8 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C2×C8 C2×C4 C1 # reps 1 4 2 1 4 2 2 12 4 4 4 4

Matrix representation of M5(2).19C22 in GL4(𝔽17) generated by

 6 0 15 0 10 0 0 1 1 16 11 0 10 0 3 0
,
 1 0 0 0 0 1 0 0 6 0 16 0 14 0 0 16
,
 16 0 0 0 0 1 0 0 0 0 16 0 3 0 0 1
,
 0 16 0 0 16 0 0 0 5 14 0 9 11 10 2 0
`G:=sub<GL(4,GF(17))| [6,10,1,10,0,0,16,0,15,0,11,3,0,1,0,0],[1,0,6,14,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,3,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,5,11,16,0,14,10,0,0,0,2,0,0,9,0] >;`

M5(2).19C22 in GAP, Magma, Sage, TeX

`M_5(2)._{19}C_2^2`
`% in TeX`

`G:=Group("M5(2).19C2^2");`
`// GroupNames label`

`G:=SmallGroup(128,847);`
`// by ID`

`G=gap.SmallGroup(128,847);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,2019,1411,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^9,c*a*c=a*b,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^8*c>;`
`// generators/relations`

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