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G = D4⋊3Q16  order 128 = 27

2nd semidirect product of D4 and Q16 acting via Q16/Q8=C2

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

Aliases: D4⋊3Q16, C42.205C23, Q8⋊C8⋊12C2, D4⋊C8.6C2, C4⋊C4.295D4, C4⋊2Q16⋊3C2, (C2×Q8).25D4, C4.19(C2×Q16), (C2×D4).256D4, C4.60(C4○D8), C4⋊C8.12C22, (C4×C8).26C22, D4⋊Q8.1C2, D4⋊3Q8.3C2, C4.SD16⋊5C2, C4.6Q16⋊3C2, C4⋊Q8.25C22, C4.37(C8⋊C22), (C4×D4).35C22, (C4×Q8).35C22, C2.21(D4⋊D4), C4.36(C8.C22), C22.171C22≀C2, C2.17(D4.9D4), C2.14(C22⋊Q16), (C2×C4).962(C2×D4), SmallGroup(128,376)

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

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

Generators and relations for D4⋊3Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 232 in 104 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, 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, C4⋊2Q16, D4⋊Q8, C4.SD16, D4⋊3Q8, D4⋊3Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, D4⋊D4, C22⋊Q16, D4.9D4, D4⋊3Q16

Character table of D4⋊3Q16

 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
ρ152-2-222-22-20000000000√2-√2√2-√200-√2√2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-222-20000000000-√2√2-√2√200-√2√2    symplectic lifted from Q16, Schur index 2
ρ172-2-222-22-20000000000-√2√2-√2√200√2-√2    symplectic lifted from Q16, Schur index 2
ρ182-2-22-222-20000000000√2-√2√2-√200√2-√2    symplectic lifted from Q16, Schur index 2
ρ192-22-20000-22-2i2i000000√-2-√-2-√-2√-2√2-√200    complex lifted from C4○D8
ρ202-22-20000-22-2i2i000000-√-2√-2√-2-√-2-√2√200    complex lifted from C4○D8
ρ212-22-20000-222i-2i000000√-2-√-2-√-2√-2-√2√200    complex lifted from C4○D8
ρ222-22-20000-222i-2i000000-√-2√-2√-2-√-2√2-√200    complex lifted from C4○D8
ρ234-44-400004-40000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-4000000000000002i2i-2i-2i0000    complex lifted from D4.9D4
ρ2644-4-400000000000000-2i-2i2i2i0000    complex lifted from D4.9D4

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

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

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

Matrix representation of D4⋊3Q16 ►in GL4(𝔽17) generated by

16200
16100
0010
0001
,
16200
0100
00160
00016
,
0700
12000
00011
00311
,
13800
13400
0040
00413
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,2,1,0,0,0,0,16,0,0,0,0,16],[0,12,0,0,7,0,0,0,0,0,0,3,0,0,11,11],[13,13,0,0,8,4,0,0,0,0,4,4,0,0,0,13] >;
 

D4⋊3Q16 in GAP, Magma, Sage, TeX 

D_4\rtimes_3Q_{16}
 
% in TeX
 
G:=Group("D4:3Q16");
 
// GroupNames label
 
G:=SmallGroup(128,376);
 
// by ID
 
G=gap.SmallGroup(128,376);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,184,1123,570,521,136,2804,1411,718,172]);
 
// 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^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of D4⋊3Q16 in TeX

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