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## G = C42.297C23order 128 = 27

### 158th non-split extension by C42 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.297C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C42.297C23
 Lower central C1 — C22 — C42.297C23
 Upper central C1 — C2×C4 — C42.297C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.297C23

Generators and relations for C42.297C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=ece=a2c, ede=b2d >

Subgroups: 348 in 206 conjugacy classes, 126 normal (40 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×9], Q8, C23 [×3], C23 [×2], C23 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C23×C4, C2×C4○D4, C2×C22⋊C8, C24.4C4, (C22×C8)⋊C2 [×2], C42.7C22 [×2], C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.19C24, C42.297C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ 1+4 [×2], C22.11C24, C2×C8○D4, Q8○M4(2), C42.297C23

Smallest permutation representation of C42.297C23
On 32 points
Generators in S32
```(1 19 31 10)(2 11 32 20)(3 21 25 12)(4 13 26 22)(5 23 27 14)(6 15 28 24)(7 17 29 16)(8 9 30 18)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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 10)(2 20)(3 12)(4 22)(5 14)(6 24)(7 16)(8 18)(9 30)(11 32)(13 26)(15 28)(17 29)(19 31)(21 25)(23 27)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 18)(11 20)(13 22)(15 24)(25 29)(27 31)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8○D4 2+ 1+4 Q8○M4(2) kernel C42.297C23 C2×C22⋊C8 C24.4C4 (C22×C8)⋊C2 C42.7C22 C8×D4 C8⋊9D4 C8⋊6D4 C22.19C24 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C22 C4 C2 # reps 1 1 1 2 2 2 4 2 1 4 4 4 4 8 2 2

Matrix representation of C42.297C23 in GL6(𝔽17)

 16 15 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 1 1 1 1 0 0 0 15 15 16
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 16 16 16 16 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 15 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 15 15 0 16
,
 16 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 15 15 0 16

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,15,0,0,0,0,1,15,0,0,0,0,1,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,1],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,1,15,0,0,1,0,1,15,0,0,0,0,1,0,0,0,0,0,1,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C42.297C23 in GAP, Magma, Sage, TeX

`C_4^2._{297}C_2^3`
`% in TeX`

`G:=Group("C4^2.297C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1708);`
`// by ID`

`G=gap.SmallGroup(128,1708);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,124]);`
`// Polycyclic`

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

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