Copied to
clipboard

## G = C42.302D4order 128 = 27

### 284th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.302D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.26C24 — C42.302D4
 Lower central C1 — C2 — C2×C4 — C42.302D4
 Upper central C1 — C22 — C2×C42 — C42.302D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.302D4

Generators and relations for C42.302D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=c3 >

Subgroups: 388 in 193 conjugacy classes, 88 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×2], SD16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2.D8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C41D4, C4⋊Q8 [×3], C2×M4(2) [×2], C2×SD16 [×4], C2×C4○D4, C4⋊M4(2), C4⋊SD16 [×2], D4.D4 [×2], C8⋊D4 [×4], D4⋊Q8 [×2], C4.Q16 [×2], C22.26C24, C23.37C23, C42.302D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22, C2×C8.C22, C42.302D4

Character table of C42.302D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D size 1 1 1 1 4 8 8 2 2 2 2 2 2 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ5 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ10 1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ11 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ12 1 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ13 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ14 1 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ15 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ16 1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ17 2 2 2 2 -2 0 0 -2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 0 0 -2 2 -2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 0 2 -2 -2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 0 0 2 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 -4 4 0 0 0 0 -4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 -4 4 -4 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 4 -4 -4 4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 -4 4 -4 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ25 4 4 -4 -4 0 0 0 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 4 -4 -4 0 0 0 4 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.302D4
On 64 points
Generators in S64
```(1 59 5 63)(2 64 6 60)(3 61 7 57)(4 58 8 62)(9 36 13 40)(10 33 14 37)(11 38 15 34)(12 35 16 39)(17 26 21 30)(18 31 22 27)(19 28 23 32)(20 25 24 29)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(1 31 51 11)(2 12 52 32)(3 25 53 13)(4 14 54 26)(5 27 55 15)(6 16 56 28)(7 29 49 9)(8 10 50 30)(17 62 33 45)(18 46 34 63)(19 64 35 47)(20 48 36 57)(21 58 37 41)(22 42 38 59)(23 60 39 43)(24 44 40 61)
(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 4 5 8)(2 7 6 3)(9 16 13 12)(10 11 14 15)(17 38 21 34)(18 33 22 37)(19 36 23 40)(20 39 24 35)(25 32 29 28)(26 27 30 31)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)```

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

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

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

Matrix representation of C42.302D4 in GL8(𝔽17)

 1 0 15 0 0 0 0 0 0 0 16 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 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
,
 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 16 0 1 0 0 0 0 1 16 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 4 0 0 9 0 0 0 0 0 0 4 13 0 0 0 0 0 4 0 13 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 15 2 8 8 0 0 0 0 15 15 9 8 0 0 0 0 8 8 2 15 0 0 0 0 9 8 2 2
,
 4 0 0 9 0 0 0 0 0 0 4 13 0 0 0 0 4 13 0 13 0 0 0 0 4 0 0 13 0 0 0 0 0 0 0 0 15 2 8 8 0 0 0 0 2 2 8 9 0 0 0 0 9 9 15 2 0 0 0 0 9 8 2 2

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

C42.302D4 in GAP, Magma, Sage, TeX

`C_4^2._{302}D_4`
`% in TeX`

`G:=Group("C4^2.302D4");`
`// GroupNames label`

`G:=SmallGroup(128,1986);`
`// by ID`

`G=gap.SmallGroup(128,1986);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,675,80,4037,1027,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽