Copied to
clipboard

## G = C42.106D4order 128 = 27

### 88th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.106D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.106D4
 Lower central C1 — C2 — C2×C4 — C42.106D4
 Upper central C1 — C2×C4 — C2×C42 — C42.106D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.106D4

Generators and relations for C42.106D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 156 in 106 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×12], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C8⋊C4 [×4], C4⋊C8 [×4], C8.C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C2×C8⋊C4, C4⋊M4(2) [×2], C2×C8.C4 [×4], C42.106D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C429C4, M4(2).C4 [×2], C42.106D4

Smallest permutation representation of C42.106D4
On 64 points
Generators in S64
```(1 40 27 16)(2 37 28 13)(3 34 29 10)(4 39 30 15)(5 36 31 12)(6 33 32 9)(7 38 25 14)(8 35 26 11)(17 43 55 58)(18 48 56 63)(19 45 49 60)(20 42 50 57)(21 47 51 62)(22 44 52 59)(23 41 53 64)(24 46 54 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(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 47 3 45 5 43 7 41)(2 42 4 48 6 46 8 44)(9 50 11 56 13 54 15 52)(10 53 12 51 14 49 16 55)(17 34 23 36 21 38 19 40)(18 37 24 39 22 33 20 35)(25 64 27 62 29 60 31 58)(26 59 28 57 30 63 32 61)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 D4 D4 Q8 Q8 M4(2).C4 kernel C42.106D4 C2×C8⋊C4 C4⋊M4(2) C2×C8.C4 C8⋊C4 C42 C2×C8 C2×C8 C22×C4 C2 # reps 1 1 2 4 8 2 4 4 2 4

Matrix representation of C42.106D4 in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0
,
 16 7 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 9 0 0 0 0 9 0 0 0

`G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[16,7,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,8,0,0,0,0,8,0,0,0] >;`

C42.106D4 in GAP, Magma, Sage, TeX

`C_4^2._{106}D_4`
`% in TeX`

`G:=Group("C4^2.106D4");`
`// GroupNames label`

`G:=SmallGroup(128,581);`
`// by ID`

`G=gap.SmallGroup(128,581);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

׿
×
𝔽