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## G = C42.281D4order 128 = 27

### 263rd 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 — C42.281D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4⋊Q8 — C42.281D4
 Lower central C1 — C2 — C2×C4 — C42.281D4
 Upper central C1 — C22 — C2×C42 — C42.281D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.281D4

Generators and relations for C42.281D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2b2c3 >

Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×14], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.12C4, Q8⋊Q8 [×4], C23.47D4 [×4], C4.SD16 [×2], C83Q8 [×2], C2×C4⋊Q8, C23.37C23, C42.281D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×SD16, C2×C8.C22, C42.281D4

Smallest permutation representation of C42.281D4
On 64 points
Generators in S64
```(1 31 17 45)(2 32 18 46)(3 25 19 47)(4 26 20 48)(5 27 21 41)(6 28 22 42)(7 29 23 43)(8 30 24 44)(9 64 50 38)(10 57 51 39)(11 58 52 40)(12 59 53 33)(13 60 54 34)(14 61 55 35)(15 62 56 36)(16 63 49 37)
(1 23 5 19)(2 4 6 8)(3 17 7 21)(9 11 13 15)(10 49 14 53)(12 51 16 55)(18 20 22 24)(25 45 29 41)(26 28 30 32)(27 47 31 43)(33 57 37 61)(34 36 38 40)(35 59 39 63)(42 44 46 48)(50 52 54 56)(58 60 62 64)
(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 54 5 50)(2 12 6 16)(3 52 7 56)(4 10 8 14)(9 17 13 21)(11 23 15 19)(18 53 22 49)(20 51 24 55)(25 58 29 62)(26 39 30 35)(27 64 31 60)(28 37 32 33)(34 41 38 45)(36 47 40 43)(42 63 46 59)(44 61 48 57)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I 4J 4K ··· 4R 8A ··· 8H order 1 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 2 ··· 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 C8.C22 2- 1+4 kernel C42.281D4 C42.12C4 Q8⋊Q8 C23.47D4 C4.SD16 C8⋊3Q8 C2×C4⋊Q8 C23.37C23 C42 C22×C4 C2×C4 C4 C4 # reps 1 1 4 4 2 2 1 1 2 2 8 2 2

Matrix representation of C42.281D4 in GL6(𝔽17)

 1 15 0 0 0 0 1 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 9 13 0 0 0 11 0 0 4
,
 0 7 0 0 0 0 5 7 0 0 0 0 0 0 10 1 0 15 0 0 16 2 2 0 0 0 6 5 15 16 0 0 5 6 1 7
,
 13 0 0 0 0 0 13 4 0 0 0 0 0 0 1 15 15 0 0 0 10 1 0 15 0 0 8 15 16 2 0 0 10 8 7 16

`G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,13,0,0,11,0,0,0,4,9,0,0,0,0,0,13,0,0,0,0,0,0,4],[0,5,0,0,0,0,7,7,0,0,0,0,0,0,10,16,6,5,0,0,1,2,5,6,0,0,0,2,15,1,0,0,15,0,16,7],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,1,10,8,10,0,0,15,1,15,8,0,0,15,0,16,7,0,0,0,15,2,16] >;`

C42.281D4 in GAP, Magma, Sage, TeX

`C_4^2._{281}D_4`
`% in TeX`

`G:=Group("C4^2.281D4");`
`// GroupNames label`

`G:=SmallGroup(128,1961);`
`// by ID`

`G=gap.SmallGroup(128,1961);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,4037,1027,124]);`
`// Polycyclic`

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

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