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## G = C42.303D4order 128 = 27

### 285th 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 — C42.303D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C23.37C23 — C42.303D4
 Lower central C1 — C2 — C2×C4 — C42.303D4
 Upper central C1 — C22 — C2×C42 — C42.303D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.303D4

Generators and relations for C42.303D4
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=a2c3 >

Subgroups: 292 in 171 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×4], C4 [×11], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×12], C23, C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], M4(2) [×2], Q16 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C2×M4(2) [×2], C2×Q16 [×4], C4⋊M4(2), C42Q16 [×4], C8.D4 [×4], Q8⋊Q8 [×4], C23.37C23 [×2], C42.303D4
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], C42.303D4

Character table of C42.303D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 8A 8B 8C 8D 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 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 -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 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 4 0 -4 0 0 0 0 0 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 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 ρ23 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 ρ24 4 4 -4 -4 0 -4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ25 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ26 4 4 -4 -4 0 4 0 0 0 -4 0 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 C42.303D4
On 64 points
Generators in S64
```(1 3 5 7)(2 8 6 4)(9 40 13 36)(10 37 14 33)(11 34 15 38)(12 39 16 35)(17 19 21 23)(18 24 22 20)(25 44 29 48)(26 41 30 45)(27 46 31 42)(28 43 32 47)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(1 54 59 21)(2 22 60 55)(3 56 61 23)(4 24 62 49)(5 50 63 17)(6 18 64 51)(7 52 57 19)(8 20 58 53)(9 41 38 28)(10 29 39 42)(11 43 40 30)(12 31 33 44)(13 45 34 32)(14 25 35 46)(15 47 36 26)(16 27 37 48)
(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 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)```

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

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

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

Matrix representation of C42.303D4 in GL8(𝔽17)

 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 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 1 0 0 0 0 0 0 0 0 1
,
 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 16 13 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 16 13 0 0 0 0 0 0 9 1
,
 0 0 9 2 0 0 0 0 0 0 10 8 0 0 0 0 11 14 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 4 16 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 1 0 0 0 0 0 0 0 4
,
 6 3 0 0 0 0 0 0 16 11 0 0 0 0 0 0 0 0 9 2 0 0 0 0 0 0 10 8 0 0 0 0 0 0 0 0 0 0 13 1 0 0 0 0 0 0 0 4 0 0 0 0 4 16 0 0 0 0 0 0 0 13 0 0

`G:=sub<GL(8,GF(17))| [16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1],[0,0,11,1,0,0,0,0,0,0,14,6,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,4],[6,16,0,0,0,0,0,0,3,11,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,13,0,0,0,0,0,0,0,1,4,0,0] >;`

C42.303D4 in GAP, Magma, Sage, TeX

`C_4^2._{303}D_4`
`% in TeX`

`G:=Group("C4^2.303D4");`
`// GroupNames label`

`G:=SmallGroup(128,1987);`
`// by ID`

`G=gap.SmallGroup(128,1987);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,352,675,80,4037,1027,124]);`
`// 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=a^2*c^3>;`
`// generators/relations`

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