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## G = C42.260D4order 128 = 27

### 242nd 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 — C42.260D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C8⋊C4 — C4×M4(2) — C42.260D4
 Lower central C1 — C2 — C2×C4 — C42.260D4
 Upper central C1 — C22 — C2×C42 — C42.260D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.260D4

Generators and relations for C42.260D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=a2b, dcd-1=c3 >

Subgroups: 340 in 185 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C422C2 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4×M4(2), SD16⋊C4 [×2], Q16⋊C4, D8⋊C4, C8⋊D4 [×2], C82D4, C8.D4, C42.78C22 [×2], C8.12D4, C8.5Q8, C23.36C23 [×2], C42.260D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D8⋊C22 [×2], C42.260D4

Smallest permutation representation of C42.260D4
On 64 points
Generators in S64
```(1 44 5 48)(2 41 6 45)(3 46 7 42)(4 43 8 47)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 40 21 36)(18 37 22 33)(19 34 23 38)(20 39 24 35)(25 57 29 61)(26 62 30 58)(27 59 31 63)(28 64 32 60)
(1 38 31 14)(2 35 32 11)(3 40 25 16)(4 37 26 13)(5 34 27 10)(6 39 28 15)(7 36 29 12)(8 33 30 9)(17 61 54 42)(18 58 55 47)(19 63 56 44)(20 60 49 41)(21 57 50 46)(22 62 51 43)(23 59 52 48)(24 64 53 45)
(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 50 5 54)(2 53 6 49)(3 56 7 52)(4 51 8 55)(9 62 13 58)(10 57 14 61)(11 60 15 64)(12 63 16 59)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)(33 43 37 47)(34 46 38 42)(35 41 39 45)(36 44 40 48)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4J 4K 4L ··· 4Q 8A ··· 8H order 1 2 2 2 2 2 2 4 ··· 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 4 8 8 2 ··· 2 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D8⋊C22 kernel C42.260D4 C4×M4(2) SD16⋊C4 Q16⋊C4 D8⋊C4 C8⋊D4 C8⋊2D4 C8.D4 C42.78C22 C8.12D4 C8.5Q8 C23.36C23 C42 C22×C4 C8 C2 # reps 1 1 2 1 1 2 1 1 2 1 1 2 2 2 8 4

Matrix representation of C42.260D4 in GL6(𝔽17)

 16 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 4 0 0 0 0 0 0 4 0 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 12 12 0 0 0 0 5 12 0 0
,
 13 8 0 0 0 0 13 4 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,0,0,0,0,12,12,0,0,5,12,0,0,0,0,5,5,0,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0] >;`

C42.260D4 in GAP, Magma, Sage, TeX

`C_4^2._{260}D_4`
`% in TeX`

`G:=Group("C4^2.260D4");`
`// GroupNames label`

`G:=SmallGroup(128,1915);`
`// by ID`

`G=gap.SmallGroup(128,1915);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,184,521,80,4037,124]);`
`// 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^-1*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;`
`// generators/relations`

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