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## G = D8.10D4order 128 = 27

### 2nd non-split extension by D8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D8.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4○D8 — D8.10D4
 Lower central C1 — C2 — C4 — C2×C8 — D8.10D4
 Upper central C1 — C22 — C22×C4 — C22×C8 — D8.10D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.10D4

Generators and relations for D8.10D4
G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 276 in 110 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C8 [×2], C8, C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16 [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×6], Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×2], Q32 [×2], C22⋊Q8, C22×C8, C2×D8, C2×SD16, C2×Q16 [×2], C4○D8 [×4], C2×C4○D4, C22⋊C16, C2.D16, C2.Q32, C8.18D4, C2×SD32, C2×Q32, C2×C4○D8, D8.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C4○D16, Q32⋊C2, D8.10D4

Character table of D8.10D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 4 8 8 2 2 2 2 8 8 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ9 2 2 2 2 -2 0 0 2 -2 -2 2 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 2 0 0 -2 2 -2 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 0 0 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 0 -2 2 2 0 0 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 0 0 0 2 0 0 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 -2 2 0 2 -2 2 0 0 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ16 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ17 2 2 2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 -√2 -√2 √2 √2 √2 orthogonal lifted from D8 ρ18 2 2 2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 √2 √2 -√2 -√2 -√2 orthogonal lifted from D8 ρ19 2 -2 2 -2 0 0 0 0 2i -2i 0 0 0 0 0 -√2 √2 √2 -√2 √-2 -√-2 ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 -ζ165+ζ163 complex lifted from C4○D16 ρ20 2 -2 2 -2 0 0 0 0 -2i 2i 0 0 0 0 0 -√2 √2 √2 -√2 -√-2 √-2 ζ1613+ζ1611 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 -ζ165+ζ163 complex lifted from C4○D16 ρ21 2 -2 2 -2 0 0 0 0 2i -2i 0 0 0 0 0 -√2 √2 √2 -√2 √-2 -√-2 ζ1613+ζ1611 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ165-ζ163 complex lifted from C4○D16 ρ22 2 -2 2 -2 0 0 0 0 -2i 2i 0 0 0 0 0 -√2 √2 √2 -√2 -√-2 √-2 ζ165+ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ165-ζ163 complex lifted from C4○D16 ρ23 2 -2 2 -2 0 0 0 0 -2i 2i 0 0 0 0 0 √2 -√2 -√2 √2 √-2 -√-2 ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 -ζ167+ζ16 complex lifted from C4○D16 ρ24 2 -2 2 -2 0 0 0 0 2i -2i 0 0 0 0 0 √2 -√2 -√2 √2 -√-2 √-2 ζ1615+ζ169 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 -ζ167+ζ16 complex lifted from C4○D16 ρ25 2 -2 2 -2 0 0 0 0 -2i 2i 0 0 0 0 0 √2 -√2 -√2 √2 √-2 -√-2 ζ1615+ζ169 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ167-ζ16 complex lifted from C4○D16 ρ26 2 -2 2 -2 0 0 0 0 2i -2i 0 0 0 0 0 √2 -√2 -√2 √2 -√-2 √-2 ζ167+ζ16 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ167-ζ16 complex lifted from C4○D16 ρ27 4 -4 -4 4 0 0 0 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of D8.10D4
On 64 points
Generators in S64
```(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 40)(34 39)(35 38)(36 37)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)
(1 16 19 64)(2 15 20 63)(3 14 21 62)(4 13 22 61)(5 12 23 60)(6 11 24 59)(7 10 17 58)(8 9 18 57)(25 52 36 41)(26 51 37 48)(27 50 38 47)(28 49 39 46)(29 56 40 45)(30 55 33 44)(31 54 34 43)(32 53 35 42)
(1 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 40 13 36)(10 39 14 35)(11 38 15 34)(12 37 16 33)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 57 29 61)(26 64 30 60)(27 63 31 59)(28 62 32 58)```

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

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

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

Matrix representation of D8.10D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 0 11 0 0 3 11
,
 16 0 0 0 0 1 0 0 0 0 0 11 0 0 14 0
,
 0 1 0 0 16 0 0 0 0 0 10 5 0 0 4 7
,
 0 1 0 0 1 0 0 0 0 0 11 14 0 0 1 6
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,1,0,0,0,0,0,14,0,0,11,0],[0,16,0,0,1,0,0,0,0,0,10,4,0,0,5,7],[0,1,0,0,1,0,0,0,0,0,11,1,0,0,14,6] >;`

D8.10D4 in GAP, Magma, Sage, TeX

`D_8._{10}D_4`
`% in TeX`

`G:=Group("D8.10D4");`
`// GroupNames label`

`G:=SmallGroup(128,921);`
`// by ID`

`G=gap.SmallGroup(128,921);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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