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G = Q8⋊2SD16  order 128 = 27

1st semidirect product of Q8 and SD16 acting via SD16/D4=C2

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

Aliases: Q8⋊2SD16, C42.192C23, D4⋊C8⋊12C2, Q8⋊C8⋊22C2, C4⋊C4.51D4, C8⋊5D4⋊15C2, C4.Q16⋊2C2, (C2×D4).47D4, C4.D8⋊7C2, C4⋊C8.6C22, D4⋊2Q8⋊30C2, C4.56(C4○D8), (C2×Q8).196D4, Q8⋊6D4.1C2, C4.28(C2×SD16), C4⋊Q8.13C22, C4.34(C8⋊C22), (C4×C8).244C22, (C4×D4).25C22, (C4×Q8).25C22, C2.20(D4⋊4D4), C4⋊1D4.16C22, C4.60(C8.C22), C2.13(D4.7D4), C22.158C22≀C2, C2.11(C22⋊SD16), (C2×C4).949(C2×D4), SmallGroup(128,363)

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

Derived series
C1 — C42 — Q8⋊2SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — Q8⋊6D4 — Q8⋊2SD16
Lower central
C1 — C22 — C42 — Q8⋊2SD16
Upper central
C1 — C22 — C42 — Q8⋊2SD16
Jennings
C1 — C22 — C22 — C42 — Q8⋊2SD16

Generators and relations for Q8⋊2SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a-1b, bd=db, dcd=c3 >

Subgroups: 336 in 128 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×SD16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, D4⋊2Q8, C4.Q16, C8⋊5D4, Q8⋊6D4, Q8⋊2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C22⋊SD16, D4.7D4, D4⋊4D4, Q8⋊2SD16

Character table of Q8⋊2SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111188822224444481644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1111111-11-111-1-1-1-11-11-1    linear of order 2
ρ41111-1-1-1111111-11-11-11111-11-11    linear of order 2
ρ51111-11111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111-11111111-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ711111-1-111111-11-11-1-111111-11-1    linear of order 2
ρ811111-1-111111-11-11-11-1-1-1-1-11-11    linear of order 2
ρ922222002-22-2-20-20-20000000000    orthogonal lifted from D4
ρ102222000-22-22-22020-2000000000    orthogonal lifted from D4
ρ112222-2002-22-2-202020000000000    orthogonal lifted from D4
ρ1222220-22-2-2-2-2200000000000000    orthogonal lifted from D4
ρ132222000-22-22-2-20-202000000000    orthogonal lifted from D4
ρ14222202-2-2-2-2-2200000000000000    orthogonal lifted from D4
ρ1522-2-200020-2000-2i02i00√-2√-2-√-2-√-2-√20√20    complex lifted from C4○D8
ρ1622-2-200020-20002i0-2i00-√-2-√-2√-2√-2-√20√20    complex lifted from C4○D8
ρ1722-2-200020-20002i0-2i00√-2√-2-√-2-√-2√20-√20    complex lifted from C4○D8
ρ182-22-2000020-20-202000√-2-√-2√-2-√-20√-20-√-2    complex lifted from SD16
ρ192-22-2000020-2020-2000√-2-√-2√-2-√-20-√-20√-2    complex lifted from SD16
ρ2022-2-200020-2000-2i02i00-√-2-√-2√-2√-2√20-√20    complex lifted from C4○D8
ρ212-22-2000020-20-202000-√-2√-2-√-2√-20-√-20√-2    complex lifted from SD16
ρ222-22-2000020-2020-2000-√-2√-2-√-2√-20√-20-√-2    complex lifted from SD16
ρ234-4-44000000000000002-2-220000    orthogonal lifted from D4⋊4D4
ρ244-44-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-4400000000000000-222-20000    orthogonal lifted from D4⋊4D4
ρ2644-4-4000-4040000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of Q8⋊2SD16
►On 64 points
Generators in S64
(1 20 57 38)(2 21 58 39)(3 22 59 40)(4 23 60 33)(5 24 61 34)(6 17 62 35)(7 18 63 36)(8 19 64 37)(9 50 45 28)(10 51 46 29)(11 52 47 30)(12 53 48 31)(13 54 41 32)(14 55 42 25)(15 56 43 26)(16 49 44 27)

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

(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)

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

Matrix representation of Q8⋊2SD16 ►in GL4(𝔽17) generated by

0100
16000
0010
0001
,
4000
01300
00160
00016
,
51200
5500
00010
001210
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,0,12,0,0,10,10],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;
 

Q8⋊2SD16 in GAP, Magma, Sage, TeX 

Q_8\rtimes_2{\rm SD}_{16}
 
% in TeX
 
G:=Group("Q8:2SD16");
 
// GroupNames label
 
G:=SmallGroup(128,363);
 
// by ID
 
G=gap.SmallGroup(128,363);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,352,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of Q8⋊2SD16 in TeX

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