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## G = C4○D4.4Q8order 128 = 27

### 2nd 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 — C2×C4 — C4○D4.4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C4○D4.4Q8
 Lower central C1 — C2 — C2×C4 — C4○D4.4Q8
 Upper central C1 — C22 — C22×C4 — C4○D4.4Q8
 Jennings C1 — C2 — C2 — C22×C4 — C4○D4.4Q8

Generators and relations for C4○D4.4Q8
G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=a-1bd2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 244 in 132 conjugacy classes, 60 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×6], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×4], C2×C8 [×7], M4(2) [×2], M4(2) [×9], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C8.C4 [×2], C2×C4⋊C4 [×2], C22×C8, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, C22.C42 [×2], C23.36D4 [×2], C2×C4.Q8, C2×C8.C4, C2×C8○D4, C4○D4.4Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, D4.3D4 [×2], C4○D4.4Q8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q8 C4○D4 C4○D4 D4.3D4 kernel C4○D4.4Q8 C22.C42 C23.36D4 C2×C4.Q8 C2×C8.C4 C2×C8○D4 C8○D4 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 4 1 1 2 2 2 4

Matrix representation of C4○D4.4Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1
,
 13 0 0 0 0 0 11 4 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 12 1 0 0 0 0 8 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[13,11,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[12,8,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C4○D4.4Q8 in GAP, Magma, Sage, TeX

`C_4\circ D_4._4Q_8`
`% in TeX`

`G:=Group("C4oD4.4Q8");`
`// GroupNames label`

`G:=SmallGroup(128,547);`
`// by ID`

`G=gap.SmallGroup(128,547);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,718,172,1027,124]);`
`// Polycyclic`

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

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