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## G = C23.5C42order 128 = 27

### 5th non-split extension by C23 of C42 acting via C42/C4=C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.5C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — C23.5C42
 Lower central C1 — C2 — C22 — C23.5C42
 Upper central C1 — C8 — C22×C8 — C23.5C42
 Jennings C1 — C2 — C2 — C22×C4 — C23.5C42

Generators and relations for C23.5C42
G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 228 in 142 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, M4(2)⋊4C4, C82M4(2), C23.C23, M4(2).8C22, C2×C8○D4, C23.5C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C23.5C42

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4O 8A 8B 8C 8D 8E ··· 8J 8K ··· 8V order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 8 ··· 8 size 1 1 2 2 2 4 4 1 1 2 2 2 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 D4 C4○D4 C23.5C42 kernel C23.5C42 M4(2)⋊4C4 C8○2M4(2) C23.C23 M4(2).8C22 C2×C8○D4 C23⋊C4 C4.D4 C4.10D4 C22×C8 C2×M4(2) C2×C8 C2×C4 C1 # reps 1 2 2 1 1 1 8 4 4 4 4 4 4 4

Matrix representation of C23.5C42 in GL4(𝔽17) generated by

 0 4 0 0 13 0 0 0 16 1 4 9 16 0 4 13
,
 1 0 0 0 0 1 0 0 0 0 16 0 4 13 0 16
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 0 1 0 4 13 1 15 0 1 0 0 0 0 0 4
,
 0 2 0 0 15 0 0 0 8 9 2 13 8 0 2 15
`G:=sub<GL(4,GF(17))| [0,13,16,16,4,0,1,0,0,0,4,4,0,0,9,13],[1,0,0,4,0,1,0,13,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,0,13,1,0,1,1,0,0,0,15,0,4],[0,15,8,8,2,0,9,0,0,0,2,2,0,0,13,15] >;`

C23.5C42 in GAP, Magma, Sage, TeX

`C_2^3._5C_4^2`
`% in TeX`

`G:=Group("C2^3.5C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,489);`
`// by ID`

`G=gap.SmallGroup(128,489);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,172,4037]);`
`// Polycyclic`

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

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