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## G = M4(2).6Q8order 128 = 27

### 4th 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).6Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).6Q8
 Lower central C1 — C2 — C2×C4 — M4(2).6Q8
 Upper central C1 — C22 — C22×C4 — M4(2).6Q8
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).6Q8

Generators and relations for M4(2).6Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a5b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 172 in 98 conjugacy classes, 56 normal (30 characteristic)
C1, C2 [×3], 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 [×2], C2.D8 [×4], C8.C4 [×4], C2×C4⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C22.C42 [×2], C82M4(2), C2×C2.D8, M4(2)⋊C4 [×2], C2×C8.C4, M4(2).6Q8
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, D4.4D4, D4.5D4, M4(2).6Q8

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

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

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

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

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 4 type + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 C4○D4 C4○D4 D4.4D4 D4.5D4 kernel M4(2).6Q8 C22.C42 C8○2M4(2) C2×C2.D8 M4(2)⋊C4 C2×C8.C4 C8.C4 C4⋊C4 C2×C8 C2×C8 M4(2) C2×C4 C23 C2 C2 # reps 1 2 1 1 2 1 8 2 2 2 2 2 2 2 2

Matrix representation of M4(2).6Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 3 14 0 0 0 0 3 3 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 13 0 0 0 0 0 0 13 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 9 0 0 0 0 15 0 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 0 0 12 5 0 0 0 0 5 5

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,3,3,0,0,0,0,14,3,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[0,15,0,0,0,0,9,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,12,5,0,0,0,0,5,5] >;`

M4(2).6Q8 in GAP, Magma, Sage, TeX

`M_4(2)._6Q_8`
`% in TeX`

`G:=Group("M4(2).6Q8");`
`// GroupNames label`

`G:=SmallGroup(128,684);`
`// by ID`

`G=gap.SmallGroup(128,684);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2019,1018,248,2804,172]);`
`// 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^5*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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