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G = D4.3Q16  order 128 = 27

The non-split extension by D4 of Q16 acting via Q16/Q8=C2

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

Aliases: D4.3Q16, Q8.4SD16, C42.198C23, Q8⋊C8⋊23C2, D4⋊C8.4C2, C4⋊C4.28D4, (D4×Q8).1C2, C4.Q16⋊4C2, C4⋊C8.9C22, C4.18(C2×Q16), (C2×D4).253D4, (C4×C8).20C22, (C2×Q8).198D4, D4⋊2Q8.5C2, C4.SD16⋊2C2, C4.30(C2×SD16), C4⋊Q8.19C22, C4.10D8⋊10C2, C4.36(C8⋊C22), (C4×D4).29C22, (C4×Q8).29C22, C4.35(C8.C22), C2.18(D4.8D4), C22.164C22≀C2, C2.13(C22⋊SD16), C2.13(C22⋊Q16), (C2×C4).955(C2×D4), SmallGroup(128,369)

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×Q8 — 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=a2c4, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 256 in 114 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, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, D4⋊2Q8, C4.Q16, C4.SD16, D4×Q8, D4.3Q16
Quotients: C1, C2, C22, D4, C23, SD16, Q16, C2×D4, C22≀C2, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C22⋊SD16, C22⋊Q16, D4.8D4, 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
ρ162-2-222-22-20000000000-√2√2-√2√200√2-√2    symplectic lifted from Q16, Schur index 2
ρ172-2-22-222-20000000000-√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
ρ192-22-20000-22-22000000√-2-√-2-√-2√-2-√-2√-200    complex lifted from SD16
ρ202-22-20000-222-2000000√-2-√-2-√-2√-2√-2-√-200    complex lifted from SD16
ρ212-22-20000-22-22000000-√-2√-2√-2-√-2√-2-√-200    complex lifted from SD16
ρ222-22-20000-222-2000000-√-2√-2√-2-√-2-√-2√-200    complex lifted from SD16
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 D4.8D4
ρ2644-4-400000000000000-2i-2i2i2i0000    complex lifted from D4.8D4

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

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

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

0100
16000
0010
0001
,
0100
1000
00160
00016
,
51200
5500
0020
0059
,
1000
01600
00162
00161
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,2,5,0,0,0,9],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,2,1] >;
 

D4.3Q16 in GAP, Magma, Sage, TeX 

D_4._3Q_{16}
 
% in TeX
 
G:=Group("D4.3Q16");
 
// GroupNames label
 
G:=SmallGroup(128,369);
 
// by ID
 
G=gap.SmallGroup(128,369);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^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=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of D4.3Q16 in TeX

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