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G = C42.30D4order 128 = 27

12nd 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 — C23 — C42.30D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C42.6C4 — C42.30D4
 Lower central C1 — C22 — C23 — C42.30D4
 Upper central C1 — C22 — C2×C42 — C42.30D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.30D4

Generators and relations for C42.30D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=a-1b-1c3 >

Subgroups: 120 in 65 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, C22×C4 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C4⋊M4(2), C42.6C4 [×2], C42.30D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C22.C42, M4(2)⋊4C4 [×2], C42.30D4

Character table of C42.30D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 4 2 2 2 2 2 2 4 4 4 8 8 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 -i -i i -i -i i -1 i -1 i 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -i i -i -1 -1 1 -i -i 1 i i i linear of order 4 ρ7 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -i -i -i i i i 1 -i 1 i -1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -i -1 1 i -i i -1 i -i -i 1 i linear of order 4 ρ9 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -i -i i 1 1 -1 i -i -1 i -i i linear of order 4 ρ10 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 i -1 1 -i i -i -1 -i i i 1 -i linear of order 4 ρ11 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 i 1 -1 i -i i 1 -i -i i -1 -i linear of order 4 ρ12 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -i 1 -1 -i i -i 1 i i -i -1 i linear of order 4 ρ13 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 i i -i 1 1 -1 -i i -1 -i i -i linear of order 4 ρ14 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 i i i -i -i -i 1 i 1 -i -1 linear of order 4 ρ15 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 i -i i -1 -1 1 i i 1 -i -i -i linear of order 4 ρ16 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 i i -i i i -i -1 -i -1 -i 1 linear of order 4 ρ17 2 2 2 2 -2 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 2 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 -2 2 -2 2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 -4 4 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 -4 -4 4 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ23 4 -4 4 -4 0 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4 ρ24 4 4 -4 -4 0 0 4i 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4 ρ25 4 -4 4 -4 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4 ρ26 4 4 -4 -4 0 0 -4i 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4

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

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

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

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

Matrix representation of C42.30D4 in GL8(𝔽17)

 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 16 0 0 0 0 0 0 15 4 0 0 0 0 0 0 0 0 13 16 0 0 0 0 0 0 15 4
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13
,
 0 0 15 5 0 0 0 0 0 0 12 2 0 0 0 0 12 2 0 0 0 0 0 0 15 5 0 0 0 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 16 5 0 0 0 0 13 2 0 0 0 0 0 0 11 4 0 0
,
 15 5 0 0 0 0 0 0 12 2 0 0 0 0 0 0 0 0 15 5 0 0 0 0 0 0 12 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 13 16 0 0 0 0 0 0 15 4 0 0

`G:=sub<GL(8,GF(17))| [13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[0,0,12,15,0,0,0,0,0,0,2,5,0,0,0,0,15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,0,13,11,0,0,0,0,0,0,2,4,0,0,0,0,12,16,0,0,0,0,0,0,12,5,0,0],[15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C42.30D4 in GAP, Magma, Sage, TeX

`C_4^2._{30}D_4`
`% in TeX`

`G:=Group("C4^2.30D4");`
`// GroupNames label`

`G:=SmallGroup(128,39);`
`// by ID`

`G=gap.SmallGroup(128,39);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,248,2804,102]);`
`// Polycyclic`

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

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