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G = M4(2)○2M4(2)  order 128 = 27

Central product of M4(2) and M4(2)

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — M4(2)○2M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — M4(2)○2M4(2)
 Lower central C1 — C2 — M4(2)○2M4(2)
 Upper central C1 — C2×C4 — M4(2)○2M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)○2M4(2)

Generators and relations for M4(2)○2M4(2)
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=a4c >

Subgroups: 332 in 282 conjugacy classes, 252 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C23×C4, C4×M4(2), C82M4(2), C2×C42⋊C2, C22×M4(2), M4(2)○2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, C22×C42, Q8○M4(2), M4(2)○2M4(2)

Smallest permutation representation of M4(2)○2M4(2)
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 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)
(1 18 26 9 5 22 30 13)(2 19 27 10 6 23 31 14)(3 20 28 11 7 24 32 15)(4 21 29 12 8 17 25 16)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4Z 8A ··· 8AF order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 Q8○M4(2) kernel M4(2)○2M4(2) C4×M4(2) C8○2M4(2) C2×C42⋊C2 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C2×M4(2) C2 # reps 1 4 8 1 2 4 4 8 32 4

Matrix representation of M4(2)○2M4(2) in GL5(𝔽17)

 13 0 0 0 0 0 0 0 13 0 0 0 0 15 4 0 1 0 0 0 0 9 16 0 0
,
 16 0 0 0 0 0 1 13 0 0 0 0 16 0 0 0 0 0 1 13 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 4 1 0 0 0 0 13 0 1 13 0 0 0 0 16 0 0
,
 16 0 0 0 0 0 1 13 0 0 0 0 16 0 0 0 0 0 16 4 0 0 0 0 1

`G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,0,0,1,9,0,0,0,0,16,0,13,15,0,0,0,0,4,0,0],[16,0,0,0,0,0,1,0,0,0,0,13,16,0,0,0,0,0,1,0,0,0,0,13,16],[1,0,0,0,0,0,0,0,1,0,0,0,0,13,16,0,4,0,0,0,0,1,13,0,0],[16,0,0,0,0,0,1,0,0,0,0,13,16,0,0,0,0,0,16,0,0,0,0,4,1] >;`

M4(2)○2M4(2) in GAP, Magma, Sage, TeX

`M_4(2)\circ_2M_4(2)`
`% in TeX`

`G:=Group("M4(2)o2M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,1605);`
`// by ID`

`G=gap.SmallGroup(128,1605);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,723,2019,172]);`
`// Polycyclic`

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

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