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## G = C24.10Q8order 128 = 27

### 9th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C24.10Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.10Q8
 Lower central C1 — C2 — C2×C4 — C24.10Q8
 Upper central C1 — C2×C4 — C23×C4 — C24.10Q8
 Jennings C1 — C2 — C2 — C22×C4 — C24.10Q8

Generators and relations for C24.10Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=e2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 236 in 139 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C22 [×11], C8 [×10], C2×C4 [×4], C2×C4 [×4], C2×C4 [×10], C23 [×3], C23 [×5], C2×C8 [×4], C2×C8 [×12], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×4], C24, C22⋊C8 [×2], C22⋊C8 [×2], C22⋊C8 [×2], C8.C4 [×4], C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×5], C23×C4, C4.C42 [×2], C2×C22⋊C8, C24.4C4, C2×C8.C4 [×2], C22×M4(2), C24.10Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C8.C4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, C2×C8.C4, M4(2).C4, C24.10Q8

Smallest permutation representation of C24.10Q8
On 32 points
Generators in S32
```(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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)
(1 15 3 9 5 11 7 13)(2 31 4 25 6 27 8 29)(10 23 12 17 14 19 16 21)(18 26 20 28 22 30 24 32)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A ··· 8P 8Q 8R 8S 8T order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 2 2 4 1 1 1 1 2 2 2 2 4 4 ··· 4 8 8 8 8

38 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 C8.C4 M4(2).C4 kernel C24.10Q8 C4.C42 C2×C22⋊C8 C24.4C4 C2×C8.C4 C22×M4(2) C22⋊C8 M4(2) C22×C4 C22×C4 C24 C2×C4 C22 C2 # reps 1 2 1 1 2 1 8 4 2 1 1 4 8 2

Matrix representation of C24.10Q8 in GL4(𝔽17) generated by

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

C24.10Q8 in GAP, Magma, Sage, TeX

`C_2^4._{10}Q_8`
`% in TeX`

`G:=Group("C2^4.10Q8");`
`// GroupNames label`

`G:=SmallGroup(128,587);`
`// by ID`

`G=gap.SmallGroup(128,587);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=e^2,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;`
`// generators/relations`

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