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## G = M4(2).5Q8order 128 = 27

### 3rd non-split extension by M4(2) of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).5Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).5Q8
 Lower central C1 — C2 — C2×C4 — M4(2).5Q8
 Upper central C1 — C22 — C22×C4 — M4(2).5Q8
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).5Q8

Generators and relations for M4(2).5Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 172 in 98 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×4], C2×C8 [×3], M4(2) [×6], M4(2) [×3], C22×C4, C22×C4 [×2], C4×C8, C8⋊C4, C4.Q8 [×4], C2.D8 [×2], C8.C4 [×4], C2×C4⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C22.C42 [×2], C82M4(2), C2×C4.Q8, M4(2)⋊C4 [×2], C2×C8.C4, M4(2).5Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, D4.3D4 [×2], M4(2).5Q8

Smallest permutation representation of M4(2).5Q8
On 64 points
Generators in S64
```(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)
(1 18 43 11)(2 23 44 16)(3 20 45 13)(4 17 46 10)(5 22 47 15)(6 19 48 12)(7 24 41 9)(8 21 42 14)(25 40 55 63)(26 37 56 60)(27 34 49 57)(28 39 50 62)(29 36 51 59)(30 33 52 64)(31 38 53 61)(32 35 54 58)
(1 52 43 30)(2 49 44 27)(3 50 45 28)(4 55 46 25)(5 56 47 26)(6 53 48 31)(7 54 41 32)(8 51 42 29)(9 64 24 33)(10 57 17 34)(11 62 18 39)(12 63 19 40)(13 60 20 37)(14 61 21 38)(15 58 22 35)(16 59 23 36)```

`G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64), (1,18,43,11)(2,23,44,16)(3,20,45,13)(4,17,46,10)(5,22,47,15)(6,19,48,12)(7,24,41,9)(8,21,42,14)(25,40,55,63)(26,37,56,60)(27,34,49,57)(28,39,50,62)(29,36,51,59)(30,33,52,64)(31,38,53,61)(32,35,54,58), (1,52,43,30)(2,49,44,27)(3,50,45,28)(4,55,46,25)(5,56,47,26)(6,53,48,31)(7,54,41,32)(8,51,42,29)(9,64,24,33)(10,57,17,34)(11,62,18,39)(12,63,19,40)(13,60,20,37)(14,61,21,38)(15,58,22,35)(16,59,23,36)>;`

`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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64), (1,18,43,11)(2,23,44,16)(3,20,45,13)(4,17,46,10)(5,22,47,15)(6,19,48,12)(7,24,41,9)(8,21,42,14)(25,40,55,63)(26,37,56,60)(27,34,49,57)(28,39,50,62)(29,36,51,59)(30,33,52,64)(31,38,53,61)(32,35,54,58), (1,52,43,30)(2,49,44,27)(3,50,45,28)(4,55,46,25)(5,56,47,26)(6,53,48,31)(7,54,41,32)(8,51,42,29)(9,64,24,33)(10,57,17,34)(11,62,18,39)(12,63,19,40)(13,60,20,37)(14,61,21,38)(15,58,22,35)(16,59,23,36) );`

`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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64)], [(1,18,43,11),(2,23,44,16),(3,20,45,13),(4,17,46,10),(5,22,47,15),(6,19,48,12),(7,24,41,9),(8,21,42,14),(25,40,55,63),(26,37,56,60),(27,34,49,57),(28,39,50,62),(29,36,51,59),(30,33,52,64),(31,38,53,61),(32,35,54,58)], [(1,52,43,30),(2,49,44,27),(3,50,45,28),(4,55,46,25),(5,56,47,26),(6,53,48,31),(7,54,41,32),(8,51,42,29),(9,64,24,33),(10,57,17,34),(11,62,18,39),(12,63,19,40),(13,60,20,37),(14,61,21,38),(15,58,22,35),(16,59,23,36)])`

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 C4○D4 C4○D4 D4.3D4 kernel M4(2).5Q8 C22.C42 C8○2M4(2) C2×C4.Q8 M4(2)⋊C4 C2×C8.C4 C8.C4 C4⋊C4 C2×C8 C2×C8 M4(2) C2×C4 C23 C2 # reps 1 2 1 1 2 1 8 2 2 2 2 2 2 4

Matrix representation of M4(2).5Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 10 0 0 0 0 12 10 0 0 0 10 0 0 0 0 12 10 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 15 0 0 0 0 9 0 0 0 0 0 0 0 11 15 0 0 0 0 10 6 0 0 0 0 0 0 8 3 0 0 0 0 1 9

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,12,0,0,0,0,10,10,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,11,10,0,0,0,0,15,6,0,0,0,0,0,0,8,1,0,0,0,0,3,9] >;`

M4(2).5Q8 in GAP, Magma, Sage, TeX

`M_4(2)._5Q_8`
`% in TeX`

`G:=Group("M4(2).5Q8");`
`// GroupNames label`

`G:=SmallGroup(128,683);`
`// by ID`

`G=gap.SmallGroup(128,683);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,2804,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;`
`// generators/relations`

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