Copied to
clipboard

## G = C42.95D4order 128 = 27

### 77th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.95D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42.95D4
 Lower central C1 — C23 — C42.95D4
 Upper central C1 — C22×C4 — C42.95D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.95D4

Generators and relations for C42.95D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 268 in 156 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C22⋊C8 [×4], C4⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×4], C23×C4, C22.7C42 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C2×C42⋊C2, C42.95D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C8○D4 [×4], C23.7Q8, (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.7C22 [×2], C42.95D4

Smallest permutation representation of C42.95D4
On 64 points
Generators in S64
```(1 21 30 13)(2 10 31 18)(3 23 32 15)(4 12 25 20)(5 17 26 9)(6 14 27 22)(7 19 28 11)(8 16 29 24)(33 53 57 47)(34 44 58 50)(35 55 59 41)(36 46 60 52)(37 49 61 43)(38 48 62 54)(39 51 63 45)(40 42 64 56)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 57 13 61)(10 58 14 62)(11 59 15 63)(12 60 16 64)(17 33 21 37)(18 34 22 38)(19 35 23 39)(20 36 24 40)(25 52 29 56)(26 53 30 49)(27 54 31 50)(28 55 32 51)
(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 42 43 4 5 46 47 8)(2 3 44 45 6 7 48 41)(9 60 57 16 13 64 61 12)(10 15 58 63 14 11 62 59)(17 36 33 24 21 40 37 20)(18 23 34 39 22 19 38 35)(25 26 52 53 29 30 56 49)(27 28 54 55 31 32 50 51)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 D4 Q8 C4○D4 C8○D4 kernel C42.95D4 C22.7C42 C2×C22⋊C8 C2×C4⋊C8 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C42 C22×C4 C22×C4 C2×C4 C22 # reps 1 2 2 2 1 4 4 4 2 2 4 16

Matrix representation of C42.95D4 in GL6(𝔽17)

 13 11 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 15 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 10 8 0 0 0 0 15 7 0 0 0 0 0 0 4 15 0 0 0 0 16 13 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 7 10 0 0 0 0 2 10 0 0 0 0 0 0 4 15 0 0 0 0 0 13 0 0 0 0 0 0 15 0 0 0 0 0 9 2

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,11,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,15,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[10,15,0,0,0,0,8,7,0,0,0,0,0,0,4,16,0,0,0,0,15,13,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[7,2,0,0,0,0,10,10,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,15,9,0,0,0,0,0,2] >;`

C42.95D4 in GAP, Magma, Sage, TeX

`C_4^2._{95}D_4`
`% in TeX`

`G:=Group("C4^2.95D4");`
`// GroupNames label`

`G:=SmallGroup(128,530);`
`// by ID`

`G=gap.SmallGroup(128,530);`
`# by ID`

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

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

׿
×
𝔽