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G = C42.262C23order 128 = 27

123rd 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 — C42.262C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C2×C42⋊C2 — C42.262C23
 Lower central C1 — C22 — C42.262C23
 Upper central C1 — C2×C4 — C42.262C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.262C23

Generators and relations for C42.262C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, de=ed >

Subgroups: 284 in 201 conjugacy classes, 134 normal (36 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×8], C2×C4 [×4], C2×C4 [×12], C2×C4 [×16], C23 [×3], C23 [×5], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×4], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×2], C22⋊C8 [×6], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×2], C23×C4, C82M4(2) [×2], C2×C22⋊C8, C24.4C4, C42.6C22 [×2], C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C2×C42⋊C2, C42.262C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.262C23

Smallest permutation representation of C42.262C23
On 32 points
Generators in S32
```(1 19 31 15)(2 16 32 20)(3 21 25 9)(4 10 26 22)(5 23 27 11)(6 12 28 24)(7 17 29 13)(8 14 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)
(2 32)(4 26)(6 28)(8 30)(10 22)(12 24)(14 18)(16 20)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4○D4 C8○D4 Q8○M4(2) kernel C42.262C23 C8○2M4(2) C2×C22⋊C8 C24.4C4 C42.6C22 C42.12C4 C42.6C4 C42.7C22 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C2×C4 C22 C2 # reps 1 2 1 1 2 2 2 4 1 4 4 8 8 8 2

Matrix representation of C42.262C23 in GL4(𝔽17) generated by

 0 1 0 0 1 0 0 0 0 0 4 0 0 0 9 13
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 2 0 0 0 0 2 0 0 0 0 2 2 0 0 7 15
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 15 16
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
`G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,4,9,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,2,7,0,0,2,15],[1,0,0,0,0,1,0,0,0,0,1,15,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;`

C42.262C23 in GAP, Magma, Sage, TeX

`C_4^2._{262}C_2^3`
`% in TeX`

`G:=Group("C4^2.262C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1656);`
`// by ID`

`G=gap.SmallGroup(128,1656);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,521,124]);`
`// 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=a^-1,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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