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G = D4○SD16order 64 = 26

Central product of D4 and SD16

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

Aliases: D4SD16, Q8SD16, D4.12D4, D85C22, C8.4C23, C4.9C24, Q8.12D4, Q165C22, D4.6C23, Q8.6C23, SD166C22, M4(2)⋊7C22, 2- 1+42C2, 2+ 1+44C2, C8○D44C2, C4○D85C2, C8⋊C225C2, (C2×C8)⋊5C22, C4.42(C2×D4), (C2×SD16)⋊6C2, C4○D42C22, C8.C224C2, (C2×Q8)⋊7C22, C22.6(C2×D4), (C2×C4).44C23, C2.31(C22×D4), (C2×D4).41C22, SmallGroup(64,258)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4○SD16
C1C2C4C2×C4C4○D42+ 1+4 — D4○SD16
C1C2C4 — D4○SD16
C1C2C4○D4 — D4○SD16
C1C2C2C4 — 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 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4, D4 [×6], D4 [×9], Q8 [×2], Q8 [×3], Q8 [×3], C23 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16, SD16 [×9], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×6], C4○D4 [×4], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, D4○SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, D4○SD16

Character table of D4○SD16

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H8A8B8C8D8E
 size 1122244442222444422444
ρ11111111111111111111111    trivial
ρ2111-1-1-1-11-11-11-1-1111-1-11-11    linear of order 2
ρ3111111-1-1-1111111-11-1-1-1-1-1    linear of order 2
ρ4111-1-1-11-111-11-1-11-1111-11-1    linear of order 2
ρ511111-1-1-1-11111-1-1-1-111111    linear of order 2
ρ6111-1-111-111-11-11-1-1-1-1-11-11    linear of order 2
ρ711111-11111111-1-11-1-1-1-1-1-1    linear of order 2
ρ8111-1-11-11-11-11-11-11-111-11-1    linear of order 2
ρ911-1-1111-1-1-111-1-1-11111-1-11    linear of order 2
ρ1011-11-1-1-1-11-1-1111-111-1-1-111    linear of order 2
ρ1111-1-111-111-111-1-1-1-11-1-111-1    linear of order 2
ρ1211-11-1-111-1-1-1111-1-11111-1-1    linear of order 2
ρ1311-1-11-1-111-111-111-1-111-1-11    linear of order 2
ρ1411-11-1111-1-1-111-11-1-1-1-1-111    linear of order 2
ρ1511-1-11-11-1-1-111-1111-1-1-111-1    linear of order 2
ρ1611-11-11-1-11-1-111-111-1111-1-1    linear of order 2
ρ172222-20000-22-2-2000000000    orthogonal lifted from D4
ρ1822-2-2-2000022-22000000000    orthogonal lifted from D4
ρ1922-22200002-2-2-2000000000    orthogonal lifted from D4
ρ20222-220000-2-2-22000000000    orthogonal lifted from D4
ρ214-4000000000000000-2-22-2000    complex faithful
ρ224-40000000000000002-2-2-2000    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 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10)(2 13)(3 16)(4 11)(5 14)(6 9)(7 12)(8 15)

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

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

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

G:=TransitiveGroup(16,116);

D4○SD16 is a maximal subgroup of
D8⋊C23  SD16.A4  Q8.6S4  D4.4S4
 D8⋊D2p: D811D4  D86D4  D811D6  D86D6  D811D10  D86D10  D811D14  D86D14 ...
 D4p.D4: D8.13D4  D8○SD16  SD1613D6  C24.C23  D20.29D4  C40.C23  D28.29D4  C56.C23 ...
 D4p.C23: C8.C24  C4.C25  D4.11D12  D12.33C23  D12.34C23  D4.11D20  D20.33C23  D20.34C23 ...
D4○SD16 is a maximal quotient of
C42.275C23  C42.276C23  C42.278C23  C42.281C23  C42.16C23  C42.19C23  C42.20C23  C42.21C23  C42.23C23  C42.352C23  C42.353C23  C42.354C23  C42.355C23  C42.357C23  C42.359C23  C42.360C23  C42.366C23  C42.367C23  M4(2)⋊3Q8  C42.385C23  C42.390C23  C42.411C23  C42.423C23  C42.424C23  C42.426C23  C42.25C23  C42.27C23  C42.30C23  Q84D8  C42.501C23  Q85Q16  C42.506C23  C42.509C23  C42.510C23  C42.512C23  C42.513C23  C42.514C23  C42.517C23  Q8×SD16  SD164Q8  D84Q8  Q164Q8  SD163Q8  C42.75C23  C42.531C23
 D8⋊D2p: D810D4  D84D4  D811D6  D86D6  D811D10  D86D10  D811D14  D86D14 ...
 Q16⋊D2p: Q1610D4  Q165D4  C24.C23  C40.C23  C56.C23 ...
 SD16⋊D2p: SD16⋊D4  SD166D4  SD162D4  SD1610D4  SD1611D4  SD1613D6  D20.29D4  D28.29D4 ...
 C8pD4⋊C2: (C2×C8)⋊13D4  M4(2)⋊16D4  M4(2)⋊10D4  C42.386C23  C42.391C23  C4.2+ 1+4  C4.192+ 1+4  C42.407C23 ...
 M4(2)⋊D2p: M4(2)⋊17D4  D4.11D12  D4.11D20  D4.11D28 ...
 C4○D4⋊D2p: (C2×Q8)⋊16D4  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  C42.18C23  D12.34C23  D20.34C23  D28.34C23 ...
 (Cp×D4).D4: 2+ 1+45C4  2- 1+44C4  C4○D4.7Q8  D4.(C2×D4)  C42.15C23  (C2×C8)⋊11D4  (C2×D4).301D4  (C2×D4).302D4 ...

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

0002
0010
0200
1000
,
2000
0200
0010
0001
,
0100
1100
0012
0020
,
2200
0100
0010
0012
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

Export

Character table of D4○SD16 in TeX

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