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G = D8○SD16  order 128 = 27

Central product of D8 and SD16

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

Aliases: D8○SD16, SD16○Q16, D8.8D4, Q16.13D4, SD16.3D4, C42.455C23, M4(2).15C23, 2- 1+4⋊4C22, C22.32+ 1+4, 2+ 1+4.7C22, C2.74D42, C8○D8⋊8C2, D4○D8⋊3C2, Q8○D8⋊3C2, C8.11(C2×D4), (C4×C8)⋊33C22, D4○SD16⋊3C2, D4.33(C2×D4), C8○D4⋊4C22, C4≀C2⋊12C22, Q8.33(C2×D4), D4.9D4⋊7C2, D4.5D4⋊6C2, D4.3D4⋊5C2, D4.4D4⋊6C2, D4.8D4⋊7C2, C8⋊C22⋊2C22, (C2×C4).23C24, (C2×D4).9C23, (C2×Q8).7C23, C8.12D4⋊24C2, (C2×C8).287C23, (C2×Q16)⋊31C22, C4○D8.27C22, C4○D4.12C23, (C2×D8).82C22, C4.D4⋊3C22, C4.104(C22×D4), C8.C22⋊3C22, C8.C4⋊19C22, (C2×SD16)⋊31C22, C4.4D4⋊18C22, C4.10D4⋊4C22, SmallGroup(128,2022)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — D8○SD16
Chief series
C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○SD16 — D8○SD16
Lower central
C1 — C2 — C2×C4 — D8○SD16
Upper central
C1 — C2 — C2×C4 — D8○SD16
Jennings
C1 — C2 — C2 — C2×C4 — D8○SD16

Generators and relations for D8○SD16
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 476 in 227 conjugacy classes, 92 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C42, C22⋊C4, C2×C8, C2×C8, M4(2), M4(2), M4(2), D8, D8, SD16, SD16, SD16, Q16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C4×C8, C4.D4, C4.10D4, C4≀C2, C4≀C2, C8.C4, C4.4D4, C8○D4, C8○D4, C2×D8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, C8○D8, D4.8D4, D4.9D4, D4.3D4, D4.4D4, D4.5D4, C8.12D4, D4○D8, D4○SD16, Q8○D8, D8○SD16
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8○SD16

Character table of D8○SD16

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11244448822444444882222448888
ρ111111111111111111111111111111    trivial
ρ2111-1-1-1-111111-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ3111-111-1-11111-111-11-11-1-1-1-1-1-111-1-1    linear of order 2
ρ41111-1-11-111111-1-111-11-1-1-1-1-1-1-1-111    linear of order 2
ρ51111-11-1-1-11111-11-11-1-11111111-1-11    linear of order 2
ρ6111-11-11-1-1111-11-111-1-1111111-111-1    linear of order 2
ρ7111-1-1111-1111-1-11111-1-1-1-1-1-1-11-11-1    linear of order 2
ρ811111-1-11-111111-1-111-1-1-1-1-1-1-1-11-11    linear of order 2
ρ9111-1-1-11-1-111-1-1-1-11-111-1-111-1111-11    linear of order 2
ρ10111111-1-1-111-1111-1-111-1-111-11-1-11-1    linear of order 2
ρ111111-1-1-11-111-11-1-1-1-1-1111-1-11-1111-1    linear of order 2
ρ12111-11111-111-1-1111-1-1111-1-11-1-1-1-11    linear of order 2
ρ13111-11-1-11111-1-11-1-1-1-1-1-1-111-111-111    linear of order 2
ρ141111-1111111-11-111-1-1-1-1-111-11-11-1-1    linear of order 2
ρ1511111-11-1111-111-11-11-111-1-11-11-1-1-1    linear of order 2
ρ16111-1-11-1-1111-1-1-11-1-11-111-1-11-1-1111    linear of order 2
ρ1722-20-20-2002-200202000-2-200200000    orthogonal lifted from D4
ρ1822-2020-2002-200-2020002200-200000    orthogonal lifted from D4
ρ1922-2-202000-22020-2000000-2-2020000    orthogonal lifted from D4
ρ2022-220-2000-220-202000000-2-2020000    orthogonal lifted from D4
ρ2122-2-20-2000-220202000000220-20000    orthogonal lifted from D4
ρ2222-20-202002-20020-20002200-200000    orthogonal lifted from D4
ρ2322-2202000-220-20-2000000220-20000    orthogonal lifted from D4
ρ2422-20202002-200-20-2000-2-200200000    orthogonal lifted from D4
ρ25444000000-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4000000000-2i00002i002√-2-2√-2-2√22√2000000    complex faithful
ρ274-4000000000-2i00002i00-2√-22√-22√2-2√2000000    complex faithful
ρ284-40000000002i0000-2i00-2√-22√-2-2√22√2000000    complex faithful
ρ294-40000000002i0000-2i002√-2-2√-22√2-2√2000000    complex faithful

Smallest permutation representation of D8○SD16
►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)

(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)

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

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

Matrix representation of D8○SD16 ►in GL4(𝔽17) generated by

141400
31400
1401414
03314
,
1000
01600
0110
160016
,
125100
512010
1201212
0121212
,
16000
01600
0110
1001
G:=sub<GL(4,GF(17))| [14,3,14,0,14,14,0,3,0,0,14,3,0,0,14,14],[1,0,0,16,0,16,1,0,0,0,1,0,0,0,0,16],[12,5,12,0,5,12,0,12,10,0,12,12,0,10,12,12],[16,0,0,1,0,16,1,0,0,0,1,0,0,0,0,1] >;
 

D8○SD16 in GAP, Magma, Sage, TeX 

D_8\circ {\rm SD}_{16}
 
% in TeX
 
G:=Group("D8oSD16");
 
// GroupNames label
 
G:=SmallGroup(128,2022);
 
// by ID
 
G=gap.SmallGroup(128,2022);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,346,2804,1411,375,172,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,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 D8○SD16 in TeX

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