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G = D4⋊2SD16  order 128 = 27

1st semidirect product of D4 and SD16 acting via SD16/D4=C2

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

Aliases: D4⋊2SD16, C42.190C23, D42.1C2, D4⋊C8⋊18C2, C4⋊C4.50D4, C4⋊Q8⋊1C22, C8⋊5D4⋊14C2, C4⋊C8⋊42C22, (C4×C8)⋊39C22, D4⋊2Q8⋊29C2, (C2×D4).251D4, C4.D8⋊12C2, C4.27(C2×SD16), C4.33(C8⋊C22), (C4×D4).24C22, C2.18(D4⋊4D4), C4⋊1D4.14C22, C22.156C22≀C2, C2.10(C22⋊SD16), (C2×C4).947(C2×D4), SmallGroup(128,361)

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

Derived series
C1 — C42 — D4⋊2SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D42 — D4⋊2SD16
Lower central
C1 — C22 — C42 — D4⋊2SD16
Upper central
C1 — C22 — C42 — D4⋊2SD16
Jennings
C1 — C22 — C22 — C42 — D4⋊2SD16

Generators and relations for D4⋊2SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a-1b, dbd=a2b, dcd=c3 >

Subgroups: 472 in 157 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C4⋊Q8, C2×SD16, C22×D4, D4⋊C8, C4.D8, D4⋊2Q8, C8⋊5D4, D42, D4⋊2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D4⋊4D4, D4⋊2SD16

Character table of D4⋊2SD16

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 111144448822224881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-11-1-1-1111111-11-1-1-1-1-111-1    linear of order 2
ρ411111-11-1-1-1111111-1-111111-1-11    linear of order 2
ρ51111-1-1-1-11111111-1-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1-1-11111111-1-111111-1-1-1-1    linear of order 2
ρ71111-11-11-1-111111-11-11111-111-1    linear of order 2
ρ81111-11-11-1-111111-111-1-1-1-11-1-11    linear of order 2
ρ922220000-22-2-2-2-2200000000000    orthogonal lifted from D4
ρ10222220200022-2-2-2-20000000000    orthogonal lifted from D4
ρ1122220-20-200-2-222-202000000000    orthogonal lifted from D4
ρ122222-20-200022-2-2-220000000000    orthogonal lifted from D4
ρ132222020200-2-222-20-2000000000    orthogonal lifted from D4
ρ14222200002-2-2-2-2-2200000000000    orthogonal lifted from D4
ρ1522-2-2020-20000-220000√-2-√-2√-2-√-20-√-2√-20    complex lifted from SD16
ρ162-2-2220-20002-2000000√-2-√-2-√-2√-2-√-200√-2    complex lifted from SD16
ρ172-2-22-2020002-2000000√-2-√-2-√-2√-2√-200-√-2    complex lifted from SD16
ρ1822-2-20-2020000-220000√-2-√-2√-2-√-20√-2-√-20    complex lifted from SD16
ρ192-2-2220-20002-2000000-√-2√-2√-2-√-2√-200-√-2    complex lifted from SD16
ρ202-2-22-2020002-2000000-√-2√-2√-2-√-2-√-200√-2    complex lifted from SD16
ρ2122-2-2020-20000-220000-√-2√-2-√-2√-20√-2-√-20    complex lifted from SD16
ρ2222-2-20-2020000-220000-√-2√-2-√-2√-20-√-2√-20    complex lifted from SD16
ρ2344-4-4000000004-4000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ254-44-40000000000000022-2-20000    orthogonal lifted from D4⋊4D4
ρ264-44-400000000000000-2-2220000    orthogonal lifted from D4⋊4D4

Smallest permutation representation of D4⋊2SD16
►On 32 points
Generators in S32
(1 14 25 23)(2 15 26 24)(3 16 27 17)(4 9 28 18)(5 10 29 19)(6 11 30 20)(7 12 31 21)(8 13 32 22)

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

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

Matrix representation of D4⋊2SD16 ►in GL4(𝔽17) generated by

1000
0100
0001
00160
,
16000
01600
0010
00016
,
12500
121200
00512
0055
,
16000
0100
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[12,12,0,0,5,12,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
 

D4⋊2SD16 in GAP, Magma, Sage, TeX 

D_4\rtimes_2{\rm SD}_{16}
 
% in TeX
 
G:=Group("D4:2SD16");
 
// GroupNames label
 
G:=SmallGroup(128,361);
 
// by ID
 
G=gap.SmallGroup(128,361);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d=a^2*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of D4⋊2SD16 in TeX

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