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G = D4⋊Q16  order 128 = 27

1st semidirect product of D4 and Q16 acting via Q16/Q8=C2

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

Aliases: D4⋊2Q16, C42.193C23, Q8⋊C8⋊17C2, D4⋊C8.3C2, C4⋊C4.52D4, C4.Q16⋊3C2, C4⋊Q16⋊1C2, (C2×Q8).46D4, C4.15(C2×Q16), (C2×D4).252D4, C4.32(C4○D8), (C4×C8).42C22, D4⋊3Q8.1C2, D4⋊2Q8.4C2, C4.6Q16⋊7C2, C4⋊Q8.14C22, C4⋊C8.162C22, (C4×D4).26C22, (C4×Q8).26C22, C2.21(D4⋊4D4), C4.31(C8.C22), C22.159C22≀C2, C2.14(D4.7D4), C2.10(C22⋊Q16), (C2×C4).950(C2×D4), SmallGroup(128,364)

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

Derived series
C1 — C42 — D4⋊Q16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — D4⋊Q16
Lower central
C1 — C22 — C42 — D4⋊Q16
Upper central
C1 — C22 — C42 — D4⋊Q16
Jennings
C1 — C22 — C22 — C42 — D4⋊Q16

Generators and relations for D4⋊Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 240 in 107 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×Q16, D4⋊C8, Q8⋊C8, C4.6Q16, D4⋊2Q8, C4.Q16, C4⋊Q16, D4⋊3Q8, D4⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C4○D8, C8.C22, C22⋊Q16, D4.7D4, D4⋊4D4, D4⋊Q16

Character table of D4⋊Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111144222244488881644448888
ρ111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-11-1-1-1-1-11-11    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-111111111-1-1-1-111111-11-1    linear of order 2
ρ51111-1-11111-1-11-1-111-1-1-1-1-11111    linear of order 2
ρ61111111111-1-11-11-1-1-11111-11-11    linear of order 2
ρ71111-1-11111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ81111111111-1-11-11-1-11-1-1-1-11-11-1    linear of order 2
ρ9222222-2-22200-20-200000000000    orthogonal lifted from D4
ρ10222200-2-2-2-2002002-2000000000    orthogonal lifted from D4
ρ11222200-2-2-2-200200-22000000000    orthogonal lifted from D4
ρ1222220022-2-2-2-2-22000000000000    orthogonal lifted from D4
ρ1322220022-2-222-2-2000000000000    orthogonal lifted from D4
ρ142222-2-2-2-22200-20200000000000    orthogonal lifted from D4
ρ152-22-22-200-2200000000-√2-√2√2√20√20-√2    symplectic lifted from Q16, Schur index 2
ρ162-22-22-200-2200000000√2√2-√2-√20-√20√2    symplectic lifted from Q16, Schur index 2
ρ172-22-2-2200-2200000000√2√2-√2-√20√20-√2    symplectic lifted from Q16, Schur index 2
ρ182-22-2-2200-2200000000-√2-√2√2√20-√20√2    symplectic lifted from Q16, Schur index 2
ρ192-2-22002-200-2i2i000000-√2√2√2-√2√-20-√-20    complex lifted from C4○D8
ρ202-2-22002-200-2i2i000000√2-√2-√2√2-√-20√-20    complex lifted from C4○D8
ρ212-2-22002-2002i-2i000000-√2√2√2-√2-√-20√-20    complex lifted from C4○D8
ρ222-2-22002-2002i-2i000000√2-√2-√2√2√-20-√-20    complex lifted from C4○D8
ρ2344-4-400000000000000-22-220000    orthogonal lifted from D4⋊4D4
ρ2444-4-4000000000000002-22-20000    orthogonal lifted from D4⋊4D4
ρ254-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-400004-40000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D4⋊Q16
►On 64 points
Generators in S64
(1 60 10 19)(2 61 11 20)(3 62 12 21)(4 63 13 22)(5 64 14 23)(6 57 15 24)(7 58 16 17)(8 59 9 18)(25 56 39 47)(26 49 40 48)(27 50 33 41)(28 51 34 42)(29 52 35 43)(30 53 36 44)(31 54 37 45)(32 55 38 46)

(1 23)(2 6)(3 58)(4 9)(5 19)(7 62)(8 13)(10 64)(11 15)(12 17)(14 60)(16 21)(18 22)(20 57)(24 61)(25 52)(26 36)(27 45)(28 32)(29 56)(30 40)(31 41)(33 54)(34 38)(35 47)(37 50)(39 43)(42 55)(44 48)(46 51)(49 53)(59 63)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 33 5 37)(2 40 6 36)(3 39 7 35)(4 38 8 34)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(17 43 21 47)(18 42 22 46)(19 41 23 45)(20 48 24 44)(49 57 53 61)(50 64 54 60)(51 63 55 59)(52 62 56 58)
 
G:=sub<Sym(64)| (1,60,10,19)(2,61,11,20)(3,62,12,21)(4,63,13,22)(5,64,14,23)(6,57,15,24)(7,58,16,17)(8,59,9,18)(25,56,39,47)(26,49,40,48)(27,50,33,41)(28,51,34,42)(29,52,35,43)(30,53,36,44)(31,54,37,45)(32,55,38,46), (1,23)(2,6)(3,58)(4,9)(5,19)(7,62)(8,13)(10,64)(11,15)(12,17)(14,60)(16,21)(18,22)(20,57)(24,61)(25,52)(26,36)(27,45)(28,32)(29,56)(30,40)(31,41)(33,54)(34,38)(35,47)(37,50)(39,43)(42,55)(44,48)(46,51)(49,53)(59,63), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,43,21,47)(18,42,22,46)(19,41,23,45)(20,48,24,44)(49,57,53,61)(50,64,54,60)(51,63,55,59)(52,62,56,58)>;
 
G:=Group( (1,60,10,19)(2,61,11,20)(3,62,12,21)(4,63,13,22)(5,64,14,23)(6,57,15,24)(7,58,16,17)(8,59,9,18)(25,56,39,47)(26,49,40,48)(27,50,33,41)(28,51,34,42)(29,52,35,43)(30,53,36,44)(31,54,37,45)(32,55,38,46), (1,23)(2,6)(3,58)(4,9)(5,19)(7,62)(8,13)(10,64)(11,15)(12,17)(14,60)(16,21)(18,22)(20,57)(24,61)(25,52)(26,36)(27,45)(28,32)(29,56)(30,40)(31,41)(33,54)(34,38)(35,47)(37,50)(39,43)(42,55)(44,48)(46,51)(49,53)(59,63), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,43,21,47)(18,42,22,46)(19,41,23,45)(20,48,24,44)(49,57,53,61)(50,64,54,60)(51,63,55,59)(52,62,56,58) );
 
G=PermutationGroup([[(1,60,10,19),(2,61,11,20),(3,62,12,21),(4,63,13,22),(5,64,14,23),(6,57,15,24),(7,58,16,17),(8,59,9,18),(25,56,39,47),(26,49,40,48),(27,50,33,41),(28,51,34,42),(29,52,35,43),(30,53,36,44),(31,54,37,45),(32,55,38,46)], [(1,23),(2,6),(3,58),(4,9),(5,19),(7,62),(8,13),(10,64),(11,15),(12,17),(14,60),(16,21),(18,22),(20,57),(24,61),(25,52),(26,36),(27,45),(28,32),(29,56),(30,40),(31,41),(33,54),(34,38),(35,47),(37,50),(39,43),(42,55),(44,48),(46,51),(49,53),(59,63)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,33,5,37),(2,40,6,36),(3,39,7,35),(4,38,8,34),(9,28,13,32),(10,27,14,31),(11,26,15,30),(12,25,16,29),(17,43,21,47),(18,42,22,46),(19,41,23,45),(20,48,24,44),(49,57,53,61),(50,64,54,60),(51,63,55,59),(52,62,56,58)]])
 

Matrix representation of D4⋊Q16 ►in GL4(𝔽17) generated by

1000
0100
0001
00160
,
16000
01600
0001
0010
,
31400
3300
00143
001414
,
5500
51200
0040
00013
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[3,3,0,0,14,3,0,0,0,0,14,14,0,0,3,14],[5,5,0,0,5,12,0,0,0,0,4,0,0,0,0,13] >;
 

D4⋊Q16 in GAP, Magma, Sage, TeX 

D_4\rtimes Q_{16}
 
% in TeX
 
G:=Group("D4:Q16");
 
// GroupNames label
 
G:=SmallGroup(128,364);
 
// by ID
 
G=gap.SmallGroup(128,364);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,232,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of D4⋊Q16 in TeX

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