Copied to
clipboard

## G = C42.46D4order 128 = 27

### 28th 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 — C42.46D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C23.37C23 — C42.46D4
 Lower central C1 — C2 — C2×C4 — C42.46D4
 Upper central C1 — C2×C4 — C2×C42 — C42.46D4
 Jennings C1 — C22 — C22 — C42 — C42.46D4

Generators and relations for C42.46D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 108 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×7], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×6], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C22×C8, Q8⋊C8 [×4], C2×C4⋊C8, C42.12C4, C23.37C23, C42.46D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C8○D4 [×2], C2×SD16, C2×Q16, (C22×C8)⋊C2, C2×Q8⋊C4, C42⋊C22, C42.46D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 4O 4P 8A ··· 8P order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 D4 SD16 Q16 C8○D4 C42⋊C22 kernel C42.46D4 Q8⋊C8 C2×C4⋊C8 C42.12C4 C23.37C23 C22⋊Q8 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C4 C2 # reps 1 4 1 1 1 4 4 2 2 4 4 8 2

Matrix representation of C42.46D4 in GL4(𝔽17) generated by

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

C42.46D4 in GAP, Magma, Sage, TeX

`C_4^2._{46}D_4`
`% in TeX`

`G:=Group("C4^2.46D4");`
`// GroupNames label`

`G:=SmallGroup(128,213);`
`// by ID`

`G=gap.SmallGroup(128,213);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,723,1123,570,136,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a*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