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G = C42.301D4order 128 = 27

283rd 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 — C42.301D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.26C24 — C42.301D4
 Lower central C1 — C2 — C2×C4 — C42.301D4
 Upper central C1 — C22 — C2×C42 — C42.301D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.301D4

Generators and relations for C42.301D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=ab2, cbc-1=dbd=b-1, dcd=a2c3 >

Subgroups: 484 in 215 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], D8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×C4○D4 [×2], C4⋊M4(2), C4⋊D8 [×4], C82D4 [×4], D42Q8 [×4], C22.26C24 [×2], C42.301D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×4], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22 [×2], C42.301D4

Character table of C42.301D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D size 1 1 1 1 4 8 8 8 8 2 2 2 2 2 2 4 4 4 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 0 0 2 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 0 0 -2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 0 0 0 2 -2 -2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 0 0 0 0 -2 2 -2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 -4 4 0 0 0 0 0 0 -4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ23 4 4 -4 -4 0 0 0 0 0 -4 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 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 -4 -4 4 0 0 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.301D4
On 64 points
Generators in S64
```(1 54 5 50)(2 51 6 55)(3 56 7 52)(4 53 8 49)(9 34 13 38)(10 39 14 35)(11 36 15 40)(12 33 16 37)(17 57 21 61)(18 62 22 58)(19 59 23 63)(20 64 24 60)(25 45 29 41)(26 42 30 46)(27 47 31 43)(28 44 32 48)
(1 21 42 12)(2 13 43 22)(3 23 44 14)(4 15 45 24)(5 17 46 16)(6 9 47 18)(7 19 48 10)(8 11 41 20)(25 64 49 36)(26 37 50 57)(27 58 51 38)(28 39 52 59)(29 60 53 40)(30 33 54 61)(31 62 55 34)(32 35 56 63)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 48)(24 47)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 40)(32 39)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)```

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

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

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

Matrix representation of C42.301D4 in GL8(𝔽17)

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

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

C42.301D4 in GAP, Magma, Sage, TeX

`C_4^2._{301}D_4`
`% in TeX`

`G:=Group("C4^2.301D4");`
`// GroupNames label`

`G:=SmallGroup(128,1985);`
`// by ID`

`G=gap.SmallGroup(128,1985);`
`# by ID`

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

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

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