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## G = C2.(C8⋊7D4)  order 128 = 27

### 3rd central extension by C2 of C8⋊7D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C8⋊7D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2.(C8⋊7D4)
 Lower central C1 — C2 — C2×C4 — C2.(C8⋊7D4)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊7D4)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊7D4)

Generators and relations for C2.(C87D4)
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=b2c-1 >

Subgroups: 340 in 146 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×8], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], 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, C4×C8 [×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×C4×C8, C2×D4⋊C4 [×2], C2.(C87D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×D8, C2×SD16, C4○D8 [×2], C24.C22, C4×D8, C4×SD16, C88D4, C87D4, C4.4D8, C42.78C22, C2.(C87D4)

Smallest permutation representation of C2.(C87D4)
On 64 points
Generators in S64
```(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(1 27 55 22)(2 28 56 23)(3 29 49 24)(4 30 50 17)(5 31 51 18)(6 32 52 19)(7 25 53 20)(8 26 54 21)(9 60 40 42)(10 61 33 43)(11 62 34 44)(12 63 35 45)(13 64 36 46)(14 57 37 47)(15 58 38 48)(16 59 39 41)
(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 55)(2 8)(3 53)(4 6)(5 51)(7 49)(9 19)(10 31)(11 17)(12 29)(13 23)(14 27)(15 21)(16 25)(18 33)(20 39)(22 37)(24 35)(26 38)(28 36)(30 34)(32 40)(41 63)(42 44)(43 61)(45 59)(46 48)(47 57)(50 52)(54 56)(58 64)(60 62)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D8 SD16 C4○D4 C4○D8 kernel C2.(C8⋊7D4) C22.4Q16 C23.65C23 C24.3C22 C2×C4×C8 C2×D4⋊C4 D4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 1 1 2 8 2 2 4 4 8 8

Matrix representation of C2.(C87D4) in GL5(𝔽17)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 1 0
,
 13 0 0 0 0 0 14 3 0 0 0 14 14 0 0 0 0 0 12 12 0 0 0 5 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[13,0,0,0,0,0,14,14,0,0,0,3,14,0,0,0,0,0,12,5,0,0,0,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;`

C2.(C87D4) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes_7D_4)`
`% in TeX`

`G:=Group("C2.(C8:7D4)");`
`// GroupNames label`

`G:=SmallGroup(128,666);`
`// by ID`

`G=gap.SmallGroup(128,666);`
`# by ID`

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

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

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