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## G = D8.12D4order 128 = 27

### 4th 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 — D8.12D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — Q8○D8 — D8.12D4
 Lower central C1 — C2 — C4 — C2×C8 — D8.12D4
 Upper central C1 — C2 — C2×C4 — C8○D4 — D8.12D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.12D4

Generators and relations for D8.12D4
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c7 >

Subgroups: 240 in 103 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C16, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C4○D4, C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32, C8○D4, C2×Q16, C2×Q16, C4○D8, C4○D8, C8.C22, 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.5D4, C2×Q32, Q32⋊C2, Q8○D8, D8.12D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.12D4

Character table of D8.12D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 16A 16B 16C 16D 16E 16F size 1 1 2 4 8 2 2 4 8 8 8 16 2 2 4 8 16 4 4 4 4 8 8 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 0 2 2 -2 0 0 -2 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 -2 0 2 0 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 -2 2 -2 0 0 2 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 0 2 2 -2 0 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 2 -2 0 -2 0 2 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ16 2 2 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ17 2 2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ18 2 2 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ19 4 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 -ζ167+ζ165-ζ163+ζ16 ζ1615-ζ169-ζ165+ζ163 ζ167-ζ165+ζ163-ζ16 -ζ1615+ζ169+ζ165-ζ163 0 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 ζ167-ζ165+ζ163-ζ16 -ζ1615+ζ169+ζ165-ζ163 -ζ167+ζ165-ζ163+ζ16 ζ1615-ζ169-ζ165+ζ163 0 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 -ζ1615+ζ169+ζ165-ζ163 -ζ167+ζ165-ζ163+ζ16 ζ1615-ζ169-ζ165+ζ163 ζ167-ζ165+ζ163-ζ16 0 0 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 ζ1615-ζ169-ζ165+ζ163 ζ167-ζ165+ζ163-ζ16 -ζ1615+ζ169+ζ165-ζ163 -ζ167+ζ165-ζ163+ζ16 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D8.12D4 in GL4(𝔽17) generated by

 0 0 1 0 16 16 16 15 16 0 0 0 1 1 0 1
,
 1 11 1 2 7 0 1 0 10 9 10 3 16 7 6 6
,
 13 15 3 1 0 10 15 12 0 14 2 12 4 7 2 9
,
 0 10 15 12 13 15 3 1 13 15 10 8 4 7 2 9
`G:=sub<GL(4,GF(17))| [0,16,16,1,0,16,0,1,1,16,0,0,0,15,0,1],[1,7,10,16,11,0,9,7,1,1,10,6,2,0,3,6],[13,0,0,4,15,10,14,7,3,15,2,2,1,12,12,9],[0,13,13,4,10,15,15,7,15,3,10,2,12,1,8,9] >;`

D8.12D4 in GAP, Magma, Sage, TeX

`D_8._{12}D_4`
`% in TeX`

`G:=Group("D8.12D4");`
`// GroupNames label`

`G:=SmallGroup(128,927);`
`// by ID`

`G=gap.SmallGroup(128,927);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,2804,718,172,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^7>;`
`// generators/relations`

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