Copied to
clipboard

## G = C42.695C23order 128 = 27

### 110th non-split extension by C42 of C23 acting via C23/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C42.695C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — C42.695C23
 Lower central C1 — C2 — C42.695C23
 Upper central C1 — C2×C4 — C42.695C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.695C23

Generators and relations for C42.695C23
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 228 in 198 conjugacy classes, 174 normal (12 characteristic)
C1, C2 [×3], C2 [×2], C4 [×14], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×24], C2×C4 [×5], Q8 [×16], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×8], C22×C4, C22×C4 [×6], C2×Q8 [×12], C4×C8 [×12], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C42.12C4 [×6], C8×Q8 [×8], C2×C4×Q8, C42.695C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2- 1+4 [×2], C23.32C23, C23×C8, Q8○M4(2), C42.695C23

Smallest permutation representation of C42.695C23
On 64 points
Generators in S64
```(1 35 55 45)(2 36 56 46)(3 37 49 47)(4 38 50 48)(5 39 51 41)(6 40 52 42)(7 33 53 43)(8 34 54 44)(9 32 64 18)(10 25 57 19)(11 26 58 20)(12 27 59 21)(13 28 60 22)(14 29 61 23)(15 30 62 24)(16 31 63 17)
(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)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 31 55 17)(2 32 56 18)(3 25 49 19)(4 26 50 20)(5 27 51 21)(6 28 52 22)(7 29 53 23)(8 30 54 24)(9 46 64 36)(10 47 57 37)(11 48 58 38)(12 41 59 39)(13 42 60 40)(14 43 61 33)(15 44 62 34)(16 45 63 35)
(1 45 55 35)(2 36 56 46)(3 47 49 37)(4 38 50 48)(5 41 51 39)(6 40 52 42)(7 43 53 33)(8 34 54 44)(9 32 64 18)(10 19 57 25)(11 26 58 20)(12 21 59 27)(13 28 60 22)(14 23 61 29)(15 30 62 24)(16 17 63 31)```

`G:=sub<Sym(64)| (1,35,55,45)(2,36,56,46)(3,37,49,47)(4,38,50,48)(5,39,51,41)(6,40,52,42)(7,33,53,43)(8,34,54,44)(9,32,64,18)(10,25,57,19)(11,26,58,20)(12,27,59,21)(13,28,60,22)(14,29,61,23)(15,30,62,24)(16,31,63,17), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,17)(2,32,56,18)(3,25,49,19)(4,26,50,20)(5,27,51,21)(6,28,52,22)(7,29,53,23)(8,30,54,24)(9,46,64,36)(10,47,57,37)(11,48,58,38)(12,41,59,39)(13,42,60,40)(14,43,61,33)(15,44,62,34)(16,45,63,35), (1,45,55,35)(2,36,56,46)(3,47,49,37)(4,38,50,48)(5,41,51,39)(6,40,52,42)(7,43,53,33)(8,34,54,44)(9,32,64,18)(10,19,57,25)(11,26,58,20)(12,21,59,27)(13,28,60,22)(14,23,61,29)(15,30,62,24)(16,17,63,31)>;`

`G:=Group( (1,35,55,45)(2,36,56,46)(3,37,49,47)(4,38,50,48)(5,39,51,41)(6,40,52,42)(7,33,53,43)(8,34,54,44)(9,32,64,18)(10,25,57,19)(11,26,58,20)(12,27,59,21)(13,28,60,22)(14,29,61,23)(15,30,62,24)(16,31,63,17), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,55,17)(2,32,56,18)(3,25,49,19)(4,26,50,20)(5,27,51,21)(6,28,52,22)(7,29,53,23)(8,30,54,24)(9,46,64,36)(10,47,57,37)(11,48,58,38)(12,41,59,39)(13,42,60,40)(14,43,61,33)(15,44,62,34)(16,45,63,35), (1,45,55,35)(2,36,56,46)(3,47,49,37)(4,38,50,48)(5,41,51,39)(6,40,52,42)(7,43,53,33)(8,34,54,44)(9,32,64,18)(10,19,57,25)(11,26,58,20)(12,21,59,27)(13,28,60,22)(14,23,61,29)(15,30,62,24)(16,17,63,31) );`

`G=PermutationGroup([(1,35,55,45),(2,36,56,46),(3,37,49,47),(4,38,50,48),(5,39,51,41),(6,40,52,42),(7,33,53,43),(8,34,54,44),(9,32,64,18),(10,25,57,19),(11,26,58,20),(12,27,59,21),(13,28,60,22),(14,29,61,23),(15,30,62,24),(16,31,63,17)], [(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),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,31,55,17),(2,32,56,18),(3,25,49,19),(4,26,50,20),(5,27,51,21),(6,28,52,22),(7,29,53,23),(8,30,54,24),(9,46,64,36),(10,47,57,37),(11,48,58,38),(12,41,59,39),(13,42,60,40),(14,43,61,33),(15,44,62,34),(16,45,63,35)], [(1,45,55,35),(2,36,56,46),(3,47,49,37),(4,38,50,48),(5,41,51,39),(6,40,52,42),(7,43,53,33),(8,34,54,44),(9,32,64,18),(10,19,57,25),(11,26,58,20),(12,21,59,27),(13,28,60,22),(14,23,61,29),(15,30,62,24),(16,17,63,31)])`

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4AD 8A ··· 8AF order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 2- 1+4 Q8○M4(2) kernel C42.695C23 C42.12C4 C8×Q8 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 C2×Q8 C4 C2 # reps 1 6 8 1 6 8 2 32 2 2

Matrix representation of C42.695C23 in GL5(𝔽17)

 16 0 0 0 0 0 5 5 0 0 0 5 12 0 0 0 0 0 5 5 0 0 0 5 12
,
 13 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 9 0 0 0 0 0 2 0 15 0 0 0 2 0 15 0 0 0 15 0 0 0 0 0 15
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 12 12 0 0 0 12 5 0 0 0 7 7 5 5 0 7 10 5 12

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,5,12],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,15,0,15,0,0,0,15,0,15],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,12,12,7,7,0,12,5,7,10,0,0,0,5,5,0,0,0,5,12] >;`

C42.695C23 in GAP, Magma, Sage, TeX

`C_4^2._{695}C_2^3`
`% in TeX`

`G:=Group("C4^2.695C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1714);`
`// by ID`

`G=gap.SmallGroup(128,1714);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,124]);`
`// Polycyclic`

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

׿
×
𝔽