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G = C4○D4.5Q8order 128 = 27

3rd 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○D4.5Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C4○D4.5Q8
 Lower central C1 — C2 — C2×C4 — C4○D4.5Q8
 Upper central C1 — C22 — C22×C4 — C4○D4.5Q8
 Jennings C1 — C2 — C2 — C22×C4 — C4○D4.5Q8

Generators and relations for C4○D4.5Q8
G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=abd2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ece-1=bc, ede-1=a2d3 >

Subgroups: 244 in 132 conjugacy classes, 60 normal (30 characteristic)
C1, C2 [×3], 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], C2.D8 [×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×C2.D8, C2×C8.C4, C2×C8○D4, C4○D4.5Q8
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, D4.4D4, D4.5D4, C4○D4.5Q8

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

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

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

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

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 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q8 C4○D4 C4○D4 D4.4D4 D4.5D4 kernel C4○D4.5Q8 C22.C42 C23.36D4 C2×C2.D8 C2×C8.C4 C2×C8○D4 C8○D4 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 C2 # reps 1 2 2 1 1 1 8 4 1 1 2 2 2 2 2

Matrix representation of C4○D4.5Q8 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 8 16 13 0 0 0 8 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 15 0 0 0 0 1 1 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 1 2 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 15 0 0 0 0 1 16 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 10 3 15 0 0 0 10 0 0 15
,
 9 2 0 0 0 0 10 8 0 0 0 0 0 0 0 13 2 15 0 0 2 4 15 0 0 0 12 12 15 15 0 0 0 5 2 15

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,8,8,0,0,0,4,16,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,8,0,10,10,0,0,0,8,3,0,0,0,0,0,15,0,0,0,0,0,0,15],[9,10,0,0,0,0,2,8,0,0,0,0,0,0,0,2,12,0,0,0,13,4,12,5,0,0,2,15,15,2,0,0,15,0,15,15] >;`

C4○D4.5Q8 in GAP, Magma, Sage, TeX

`C_4\circ D_4._5Q_8`
`% in TeX`

`G:=Group("C4oD4.5Q8");`
`// GroupNames label`

`G:=SmallGroup(128,548);`
`// by ID`

`G=gap.SmallGroup(128,548);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248,718,172,1027,124]);`
`// Polycyclic`

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

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