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G = D4⋊4Q16  order 128 = 27

3rd semidirect product of D4 and Q16 acting via Q16/Q8=C2

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

Aliases: D4⋊4Q16, Q8.7D8, C42.210C23, Q8⋊C8⋊13C2, D4⋊C8.7C2, C4⋊C4.32D4, C8⋊2Q8⋊2C2, C4.31(C2×D8), (D4×Q8).2C2, C4⋊2Q16⋊4C2, C4.22(C2×Q16), (C2×D4).257D4, C4.10D8⋊2C2, C4⋊C8.15C22, (C4×C8).47C22, (C2×Q8).202D4, D4⋊Q8.2C2, C4⋊Q8.30C22, C4.67(C8⋊C22), (C4×D4).38C22, (C4×Q8).38C22, C2.17(C22⋊D8), C4.40(C8.C22), C22.176C22≀C2, C2.17(C22⋊Q16), C2.14(D4.10D4), (C2×C4).967(C2×D4), SmallGroup(128,381)

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

Derived series
C1 — C42 — D4⋊4Q16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — D4×Q8 — D4⋊4Q16
Lower central
C1 — C22 — C42 — D4⋊4Q16
Upper central
C1 — C22 — C42 — D4⋊4Q16
Jennings
C1 — C22 — C22 — C42 — D4⋊4Q16

Generators and relations for D4⋊4Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 264 in 117 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊2Q16, D4⋊Q8, C8⋊2Q8, D4×Q8, D4⋊4Q16
Quotients: C1, C2, C22, D4, C23, D8, Q16, C2×D4, C22≀C2, C2×D8, C2×Q16, C8⋊C22, C8.C22, C22⋊D8, C22⋊Q16, D4.10D4, D4⋊4Q16

Character table of D4⋊4Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111144222244488881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11111-1-11-1-111-1-1-1-1-11111    linear of order 2
ρ41111-1-11111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ51111111111-1-111-1-1-11-1-1-1-111-1-1    linear of order 2
ρ61111111111-1-111-1-1-1-11111-1-111    linear of order 2
ρ71111-1-11111111-11-1-1-1111111-1-1    linear of order 2
ρ81111-1-11111111-11-1-11-1-1-1-1-1-111    linear of order 2
ρ9222200-2-222-2-2-20200000000000    orthogonal lifted from D4
ρ10222200-2-22222-20-200000000000    orthogonal lifted from D4
ρ11222200-2-2-2-2002002-2000000000    orthogonal lifted from D4
ρ122222-2-222-2-200-22000000000000    orthogonal lifted from D4
ρ13222200-2-2-2-200200-22000000000    orthogonal lifted from D4
ρ1422222222-2-200-2-2000000000000    orthogonal lifted from D4
ρ152-22-20000-22-22000000-√2√2√2-√2√2-√200    orthogonal lifted from D8
ρ162-22-20000-222-2000000-√2√2√2-√2-√2√200    orthogonal lifted from D8
ρ172-22-20000-22-22000000√2-√2-√2√2-√2√200    orthogonal lifted from D8
ρ182-22-20000-222-2000000√2-√2-√2√2√2-√200    orthogonal lifted from D8
ρ192-2-222-22-20000000000-√2√2-√2√200√2-√2    symplectic lifted from Q16, Schur index 2
ρ202-2-222-22-20000000000√2-√2√2-√200-√2√2    symplectic lifted from Q16, Schur index 2
ρ212-2-22-222-20000000000-√2√2-√2√200-√2√2    symplectic lifted from Q16, Schur index 2
ρ222-2-22-222-20000000000√2-√2√2-√200√2-√2    symplectic lifted from Q16, Schur index 2
ρ234-44-400004-40000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-40000000000000022-2-20000    symplectic lifted from D4.10D4, Schur index 2
ρ2644-4-400000000000000-2-2220000    symplectic lifted from D4.10D4, Schur index 2

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

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

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

Matrix representation of D4⋊4Q16 ►in GL4(𝔽17) generated by

1000
0100
0012
001616
,
16000
01600
0012
00016
,
31400
3300
0006
0030
,
71600
161000
00160
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,2,16],[3,3,0,0,14,3,0,0,0,0,0,3,0,0,6,0],[7,16,0,0,16,10,0,0,0,0,16,0,0,0,0,16] >;
 

D4⋊4Q16 in GAP, Magma, Sage, TeX 

D_4\rtimes_4Q_{16}
 
% in TeX
 
G:=Group("D4:4Q16");
 
// GroupNames label
 
G:=SmallGroup(128,381);
 
// by ID
 
G=gap.SmallGroup(128,381);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of D4⋊4Q16 in TeX

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