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## G = Q8⋊D4⋊C2order 128 = 27

### 26th semidirect product of Q8⋊D4 and C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — Q8⋊D4⋊C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.31C24 — Q8⋊D4⋊C2
 Lower central C1 — C22 — C22×C4 — Q8⋊D4⋊C2
 Upper central C1 — C22 — C22×C4 — Q8⋊D4⋊C2
 Jennings C1 — C2 — C22 — C22×C4 — Q8⋊D4⋊C2

Generators and relations for Q8⋊D4⋊C2
G = < a,b,c,d,e | a4=c4=d2=e2=1, b2=a2, bab-1=cac-1=dad=a-1, eae=a-1c2, cbc-1=dbd=a-1b, ebe=abcd, dcd=c-1, ece=a2c-1, ede=a2d >

Subgroups: 364 in 141 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×SD16, C22×Q8, C2×C4○D4, C22.M4(2), C22.SD16, C23.78C23, Q8⋊D4, C22.31C24, Q8⋊D4⋊C2
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D4.10D4, C2≀C22, Q8⋊D4⋊C2

Character table of Q8⋊D4⋊C2

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D size 1 1 1 1 2 2 8 8 4 4 4 4 8 8 8 8 8 8 8 8 8 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 2 -2 -2 -2 0 -2 0 0 2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 2 2 0 0 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 -2 2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 -2 2 2 -2 0 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 2 0 -2 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 0 0 0 -2 2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ16 2 2 -2 -2 -2 2 0 0 0 2 -2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 2 -2 -2 -2 2 0 0 0 -2 2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ18 2 2 -2 -2 -2 2 0 0 0 2 -2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ19 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ21 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of Q8⋊D4⋊C2
On 32 points
Generators in S32
(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)
(1 16 3 14)(2 15 4 13)(5 28 7 26)(6 27 8 25)(9 20 11 18)(10 19 12 17)(21 32 23 30)(22 31 24 29)
(1 24 10 28)(2 23 11 27)(3 22 12 26)(4 21 9 25)(5 15 31 18)(6 14 32 17)(7 13 29 20)(8 16 30 19)
(1 22)(2 21)(3 24)(4 23)(5 20)(6 19)(7 18)(8 17)(9 27)(10 26)(11 25)(12 28)(13 31)(14 30)(15 29)(16 32)
(1 29)(2 6)(3 31)(4 8)(5 12)(7 10)(9 30)(11 32)(13 22)(14 25)(15 24)(16 27)(17 21)(18 28)(19 23)(20 26)

G:=sub<Sym(32)| (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), (1,16,3,14)(2,15,4,13)(5,28,7,26)(6,27,8,25)(9,20,11,18)(10,19,12,17)(21,32,23,30)(22,31,24,29), (1,24,10,28)(2,23,11,27)(3,22,12,26)(4,21,9,25)(5,15,31,18)(6,14,32,17)(7,13,29,20)(8,16,30,19), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,27)(10,26)(11,25)(12,28)(13,31)(14,30)(15,29)(16,32), (1,29)(2,6)(3,31)(4,8)(5,12)(7,10)(9,30)(11,32)(13,22)(14,25)(15,24)(16,27)(17,21)(18,28)(19,23)(20,26)>;

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), (1,16,3,14)(2,15,4,13)(5,28,7,26)(6,27,8,25)(9,20,11,18)(10,19,12,17)(21,32,23,30)(22,31,24,29), (1,24,10,28)(2,23,11,27)(3,22,12,26)(4,21,9,25)(5,15,31,18)(6,14,32,17)(7,13,29,20)(8,16,30,19), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,27)(10,26)(11,25)(12,28)(13,31)(14,30)(15,29)(16,32), (1,29)(2,6)(3,31)(4,8)(5,12)(7,10)(9,30)(11,32)(13,22)(14,25)(15,24)(16,27)(17,21)(18,28)(19,23)(20,26) );

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)], [(1,16,3,14),(2,15,4,13),(5,28,7,26),(6,27,8,25),(9,20,11,18),(10,19,12,17),(21,32,23,30),(22,31,24,29)], [(1,24,10,28),(2,23,11,27),(3,22,12,26),(4,21,9,25),(5,15,31,18),(6,14,32,17),(7,13,29,20),(8,16,30,19)], [(1,22),(2,21),(3,24),(4,23),(5,20),(6,19),(7,18),(8,17),(9,27),(10,26),(11,25),(12,28),(13,31),(14,30),(15,29),(16,32)], [(1,29),(2,6),(3,31),(4,8),(5,12),(7,10),(9,30),(11,32),(13,22),(14,25),(15,24),(16,27),(17,21),(18,28),(19,23),(20,26)]])

Matrix representation of Q8⋊D4⋊C2 in GL6(𝔽17)

 4 0 0 0 0 0 0 13 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 16 0 1 0 0 0 16 1 0
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 2 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 1 0 16
,
 0 8 0 0 0 0 15 0 0 0 0 0 0 0 16 0 2 0 0 0 0 0 1 16 0 0 16 0 1 0 0 0 16 1 1 0
,
 0 8 0 0 0 0 15 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 16 1 0 0 0 0 16 0 0 0 0 1 16 0
,
 0 15 0 0 0 0 8 0 0 0 0 0 0 0 16 0 2 0 0 0 16 0 1 1 0 0 0 0 1 0 0 0 16 1 1 0

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

Q8⋊D4⋊C2 in GAP, Magma, Sage, TeX

Q_8\rtimes D_4\rtimes C_2
% in TeX

G:=Group("Q8:D4:C2");
// GroupNames label

G:=SmallGroup(128,336);
// by ID

G=gap.SmallGroup(128,336);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,352,1123,570,521,136,1411]);
// Polycyclic

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

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