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## G = Q8⋊D4order 64 = 26

### 1st semidirect product of Q8 and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8⋊D4
G = < a,b,c,d | a4=c4=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 145 in 79 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, Q8⋊D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8.C22, Q8⋊D4

Character table of Q8⋊D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D size 1 1 1 1 2 2 8 2 2 4 4 4 4 4 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 0 0 -2 2 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 0 0 -2 2 0 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 -2 -2 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 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 -2 2 0 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 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 2 -2 0 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 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

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

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

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

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

Q8⋊D4 is a maximal subgroup of
C24.103D4  C24.105D4  C42.226D4  C42.230D4  C42.235D4  C234SD16  C24.123D4  C24.126D4  C24.129D4  C4.152+ 1+4  C4.162+ 1+4  C42.269D4  C42.271D4  C42.276D4  Q83S4  C23.16S4  Q8⋊S4
(Cp×Q8)⋊D4: C24.178D4  (C2×Q8)⋊16D4  (C2×Q8)⋊17D4  Q83D12  D68SD16  (C3×Q8)⋊13D4  Q82D20  D108SD16 ...
C4⋊C4.D2p: C232SD16  C4⋊C4.6D4  Q8⋊D4⋊C2  C24.12D4  C42.355C23  C42.360C23  C42.407C23  C42.411C23 ...
(C2×C2p)⋊SD16: C42.223D4  C42.264D4  Dic614D4  Dic1014D4  Dic1414D4 ...
Q8⋊D4 is a maximal quotient of
C24.155D4  (C2×C4)⋊3SD16
Q8⋊D4p: Q83D8  Q83D12  Q82D20  Q82D28 ...
D2p⋊SD16: D4⋊SD16  D68SD16  D108SD16  Dic147D4 ...
(Cp×Q8)⋊D4: C23⋊SD16  C233SD16  (C3×Q8)⋊13D4  (C5×Q8)⋊13D4  (C7×Q8)⋊13D4 ...
C4⋊C4.D2p: (C2×C4)⋊SD16  C24.14D4  (C2×C4).SD16  Q8⋊SD16  Q86SD16  D4.5SD16  Q83Q16  Q84SD16 ...
C23.D4p: C23.38D8  Dic614D4  Dic1014D4  Dic1414D4 ...

Matrix representation of Q8⋊D4 in GL4(𝔽17) generated by

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

Q8⋊D4 in GAP, Magma, Sage, TeX

Q_8\rtimes D_4
% in TeX

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

G:=SmallGroup(64,129);
// by ID

G=gap.SmallGroup(64,129);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,199,362,963,489,117]);
// Polycyclic

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

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