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## G = C2.(C8⋊2D4)  order 128 = 27

### 3rd central extension by C2 of C8⋊2D4

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C8⋊2D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2.(C8⋊2D4)
 Lower central C1 — C2 — C2×C4 — C2.(C8⋊2D4)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊2D4)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊2D4)

Generators and relations for C2.(C82D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=b6c-1 >

Subgroups: 340 in 142 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×6], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, C8⋊C4 [×2], D4⋊C4 [×4], D4⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16 [×2], C23.65C23, C24.3C22, C2×C8⋊C4, C2×D4⋊C4 [×2], C2.(C82D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C8⋊C22 [×3], C8.C22, C24.C22, SD16⋊C4, D8⋊C4, C8⋊D4, C82D4, C42.28C22, C42.29C22, C2.(C82D4)

Smallest permutation representation of C2.(C82D4)
On 64 points
Generators in S64
```(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 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 52 16 33)(2 20 9 44)(3 50 10 39)(4 18 11 42)(5 56 12 37)(6 24 13 48)(7 54 14 35)(8 22 15 46)(17 63 41 31)(19 61 43 29)(21 59 45 27)(23 57 47 25)(26 53 58 34)(28 51 60 40)(30 49 62 38)(32 55 64 36)
(1 8 31 30)(2 29 32 7)(3 6 25 28)(4 27 26 5)(9 61 64 14)(10 13 57 60)(11 59 58 12)(15 63 62 16)(17 40 52 48)(18 47 53 39)(19 38 54 46)(20 45 55 37)(21 36 56 44)(22 43 49 35)(23 34 50 42)(24 41 51 33)```

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2.(C8⋊2D4) C22.4Q16 C23.65C23 C24.3C22 C2×C8⋊C4 C2×D4⋊C4 D4⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 2 1 1 1 2 8 2 2 8 3 1

Matrix representation of C2.(C82D4) in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 13 9 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 8 8 14 0 0 0 0 0 9 8 0 14 0 0 0 0 0 14 9 9 0 0 0 0 3 0 8 9
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 14 0 9 8 0 0 0 0 0 3 8 8 0 0 0 0 9 8 0 14 0 0 0 0 8 8 14 0
,
 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 8 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 8 8 14 0 0 0 0 0 8 9 0 3 0 0 0 0 14 0 9 8 0 0 0 0 0 3 8 8

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

C2.(C82D4) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes_2D_4)`
`% in TeX`

`G:=Group("C2.(C8:2D4)");`
`// GroupNames label`

`G:=SmallGroup(128,668);`
`// by ID`

`G=gap.SmallGroup(128,668);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,58,2019,248,2804,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=a*b^-1,d*c*d^-1=b^6*c^-1>;`
`// generators/relations`

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