Copied to
clipboard

## G = C42.259C23order 128 = 27

### 120th 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 — C42.259C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C42.259C23
 Lower central C1 — C22 — C42.259C23
 Upper central C1 — C2×C4 — C42.259C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.259C23

Generators and relations for C42.259C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=ab2, ae=ea, bc=cb, bd=db, be=eb, dcd=a2b2c, ece=b2c, de=ed >

Subgroups: 284 in 196 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×14], C23, C23 [×2], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C2×M4(2) [×4], C23×C4, C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C42.7C22 [×8], C2×C42⋊C2, C42.259C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, Q8○M4(2) [×2], C42.259C23

Smallest permutation representation of C42.259C23
On 32 points
Generators in S32
```(1 19 27 10)(2 15 28 24)(3 21 29 12)(4 9 30 18)(5 23 31 14)(6 11 32 20)(7 17 25 16)(8 13 26 22)
(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)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4○D4 Q8○M4(2) kernel C42.259C23 C4×M4(2) C24.4C4 C4⋊M4(2) C42.7C22 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C2×C4 C2 # reps 1 2 2 2 8 1 4 4 8 8 4

Matrix representation of C42.259C23 in GL6(𝔽17)

 13 0 0 0 0 0 13 4 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 9 0 0 0 0 4 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 4 0 0
,
 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 1 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 16 0 0 0 0 0 0 16

`G:=sub<GL(6,GF(17))| [13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C42.259C23 in GAP, Magma, Sage, TeX

`C_4^2._{259}C_2^3`
`% in TeX`

`G:=Group("C4^2.259C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1653);`
`// by ID`

`G=gap.SmallGroup(128,1653);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,100,2019,124]);`
`// 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*b^2,d*a*d=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*b^2*c,e*c*e=b^2*c,d*e=e*d>;`
`// generators/relations`

׿
×
𝔽