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## G = D4○SD16order 64 = 26

### Central product of D4 and SD16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — D4○SD16
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○SD16
 Lower central C1 — C2 — C4 — D4○SD16
 Upper central C1 — C2 — C4○D4 — D4○SD16
 Jennings C1 — C2 — C2 — C4 — D4○SD16

Generators and relations for D4○SD16
G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 205 in 129 conjugacy classes, 79 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, D4○SD16
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, D4○SD16

Character table of D4○SD16

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E size 1 1 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 2 2 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 2 2 -2 0 0 0 0 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 -2 0 0 0 0 2 2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 2 0 0 0 0 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 2 0 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 complex faithful

Permutation representations of D4○SD16
On 16 points - transitive group 16T116
Generators in S16
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)

G:=sub<Sym(16)| (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11)>;

G:=Group( (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11) );

G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,9),(3,12),(4,15),(5,10),(6,13),(7,16),(8,11)]])

G:=TransitiveGroup(16,116);

Matrix representation of D4○SD16 in GL4(𝔽3) generated by

 0 0 0 2 0 0 1 0 0 2 0 0 1 0 0 0
,
 2 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 1 0 0 0 0 1 2 0 0 2 0
,
 2 2 0 0 0 1 0 0 0 0 1 0 0 0 1 2
G:=sub<GL(4,GF(3))| [0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,1,0,0,0,0,1,2,0,0,2,0],[2,0,0,0,2,1,0,0,0,0,1,1,0,0,0,2] >;

D4○SD16 in GAP, Magma, Sage, TeX

D_4\circ {\rm SD}_{16}
% in TeX

G:=Group("D4oSD16");
// GroupNames label

G:=SmallGroup(64,258);
// by ID

G=gap.SmallGroup(64,258);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,255,1444,730,88]);
// Polycyclic

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

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