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G = SD16⋊Q8  order 128 = 27

1st semidirect product of SD16 and Q8 acting via Q8/C4=C2

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

Aliases: SD16⋊1Q8, C42.66C23, C4.1002- (1+4), (Q82)⋊7C2, C8⋊Q8⋊32C2, C8.6(C2×Q8), C2.43(D4×Q8), C4⋊C4.388D4, C8⋊4Q8⋊15C2, C8⋊2Q8⋊33C2, Q8.Q8⋊51C2, D4.11(C2×Q8), Q8.11(C2×Q8), C4.Q16⋊46C2, C2.68(D4○D8), (C2×Q8).249D4, D4⋊2Q8.2C2, (C4×SD16).5C2, C4.43(C22×Q8), C4⋊C4.274C23, C4⋊C8.144C22, (C2×C8).214C23, (C2×C4).577C24, (C4×C8).201C22, D4⋊3Q8.10C2, D4⋊Q8.12C2, C4⋊Q8.206C22, C2.D8.75C22, C8⋊C4.70C22, C4.Q8.77C22, SD16⋊C4.3C2, (C4×D4).213C22, (C2×D4).438C23, C4.53(C8.C22), (C4×Q8).204C22, (C2×Q8).411C23, D4⋊C4.94C22, C22.837(C22×D4), C42.C2.75C22, Q8⋊C4.165C22, (C2×SD16).123C22, (C2×C4).647(C2×D4), C2.90(C2×C8.C22), SmallGroup(128,2117)

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

Derived series
C1 — C2×C4 — SD16⋊Q8
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — SD16⋊Q8
Lower central
C1 — C2 — C2×C4 — SD16⋊Q8
Upper central
C1 — C22 — C4×Q8 — SD16⋊Q8
Jennings
C1 — C2 — C2 — C2×C4 — SD16⋊Q8

Subgroups: 304 in 170 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×13], C22, C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4, Q8 [×2], Q8 [×7], C23, C42, C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×2], C2×C8 [×2], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8, C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×3], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8 [×3], C2×SD16, C4×SD16, SD16⋊C4 [×2], C8⋊4Q8, D4⋊Q8 [×2], D4⋊2Q8, C4.Q16, Q8.Q8 [×2], C8⋊2Q8, C8⋊Q8 [×2], D4⋊3Q8, Q82, SD16⋊Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C8.C22 [×2], C22×D4, C22×Q8, 2- (1+4), D4×Q8, C2×C8.C22, D4○D8, SD16⋊Q8

Generators and relations
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
0051200
005500
0000125
00001212
,
100000
010000
001000
0001600
0000160
000001
,
0160000
100000
0013000
0001300
000040
000004
,
0130000
1300000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

Character table of SD16⋊Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11114422224444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111111111-11-1-11-111-1-1-1-111111-1-1    linear of order 2
ρ311111111111-111-111-11-1-111-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-1-1-11-1-111-11-1-1-1-111    linear of order 2
ρ51111-1-111111-111-111-1-1-1-1-1-1111111    linear of order 2
ρ61111-1-11111-1-1-1-1-1-11-11111-11111-1-1    linear of order 2
ρ71111-1-1111111111111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-11111-11-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ91111-1-11-11-1111-11-1-1-1-1-1111-1-111-11    linear of order 2
ρ101111-1-11-11-1-11-1111-1-111-1-11-1-1111-1    linear of order 2
ρ111111-1-11-11-11-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ121111-1-11-11-1-1-1-11-11-111-11-1111-1-1-11    linear of order 2
ρ131111111-11-11-11-1-1-1-1111-1-1-1-1-111-11    linear of order 2
ρ141111111-11-1-1-1-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ151111111-11-1111-11-1-1-11-11-1-111-1-11-1    linear of order 2
ρ161111111-11-1-11-1111-1-1-11-11-111-1-1-11    linear of order 2
ρ17222200-22-22-20220-2-2000000000000    orthogonal lifted from D4
ρ18222200-2-2-2-2-202-2022000000000000    orthogonal lifted from D4
ρ19222200-2-2-2-220-220-22000000000000    orthogonal lifted from D4
ρ20222200-22-2220-2-202-2000000000000    orthogonal lifted from D4
ρ212-22-2-22-20200200-200000000-220000    symplectic lifted from Q8, Schur index 2
ρ222-22-22-2-20200200-2000000002-20000    symplectic lifted from Q8, Schur index 2
ρ232-22-2-22-20200-2002000000002-20000    symplectic lifted from Q8, Schur index 2
ρ242-22-22-2-20200-200200000000-220000    symplectic lifted from Q8, Schur index 2
ρ2544-4-40000000000000000000002√22√200    orthogonal lifted from D4○D8
ρ2644-4-40000000000000000000002√22√200    orthogonal lifted from D4○D8
ρ274-4-44000-4040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-44-40040-400000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ294-4-4400040-40000000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

SD_{16}\rtimes Q_8
 
% in TeX
 
G:=Group("SD16:Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2117);
 
// by ID
 
G=gap.SmallGroup(128,2117);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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