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## G = C4○D4.8Q8order 128 = 27

### 6th non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4○D4.8Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C4○D4.8Q8
 Lower central C1 — C2 — C4 — C4○D4.8Q8
 Upper central C1 — C22 — C2×C4○D4 — C4○D4.8Q8
 Jennings C1 — C2 — C2 — C2×C4 — C4○D4.8Q8

Generators and relations for C4○D4.8Q8
G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=ad2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=a2d3 >

Subgroups: 356 in 238 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C2.D8, C23.25D4, M4(2)⋊C4, C23.33C23, C2×C8○D4, C4○D4.8Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, D4○D8, Q8○D8, C4○D4.8Q8

Smallest permutation representation of C4○D4.8Q8
On 64 points
Generators in S64
```(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 53 21 49)(18 54 22 50)(19 55 23 51)(20 56 24 52)(41 57 45 61)(42 58 46 62)(43 59 47 63)(44 60 48 64)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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 61 27 43)(2 60 28 42)(3 59 29 41)(4 58 30 48)(5 57 31 47)(6 64 32 46)(7 63 25 45)(8 62 26 44)(9 21 36 51)(10 20 37 50)(11 19 38 49)(12 18 39 56)(13 17 40 55)(14 24 33 54)(15 23 34 53)(16 22 35 52)```

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

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

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

44 conjugacy classes

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

44 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 Q8 D4○D8 Q8○D8 kernel C4○D4.8Q8 C2×C2.D8 C23.25D4 M4(2)⋊C4 C23.33C23 C2×C8○D4 C8○D4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 3 3 6 2 1 16 3 1 4 2 2

Matrix representation of C4○D4.8Q8 in GL6(𝔽17)

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

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

C4○D4.8Q8 in GAP, Magma, Sage, TeX

`C_4\circ D_4._8Q_8`
`% in TeX`

`G:=Group("C4oD4.8Q8");`
`// GroupNames label`

`G:=SmallGroup(128,1645);`
`// by ID`

`G=gap.SmallGroup(128,1645);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,568,521,2804,172]);`
`// Polycyclic`

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

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