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G = SD16⋊3Q8  order 128 = 27

3rd semidirect product of SD16 and Q8 acting via Q8/C4=C2

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

Aliases: SD16⋊3Q8, C42.69C23, C4.1032- (1+4), C8⋊Q8⋊34C2, C8.8(C2×Q8), C2.46(D4×Q8), C4⋊C4.391D4, C8⋊4Q8⋊17C2, Q8.Q8⋊52C2, Q8.13(C2×Q8), D4.13(C2×Q8), D4.Q8.4C2, Q8⋊3Q8⋊11C2, Q8⋊Q8⋊29C2, (C2×Q8).140D4, D4⋊2Q8.3C2, C8.5Q8⋊24C2, (C4×SD16).7C2, C4.46(C22×Q8), C4⋊C4.277C23, C4⋊C8.147C22, (C2×C8).377C23, (C2×C4).580C24, (C4×C8).204C22, D4⋊3Q8.11C2, C4⋊Q8.209C22, C8⋊C4.73C22, SD16⋊C4.5C2, (C4×D4).215C22, (C2×D4).440C23, (C2×Q8).414C23, (C4×Q8).207C22, C2.D8.141C22, C4.Q8.120C22, C2.110(D4○SD16), D4⋊C4.96C22, C22.840(C22×D4), C42.C2.78C22, Q8⋊C4.188C22, (C2×SD16).125C22, C2.105(D8⋊C22), (C2×C4).650(C2×D4), SmallGroup(128,2120)

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

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

Subgroups: 288 in 167 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C8 [×2], C8 [×3], C2×C4 [×7], C2×C4 [×11], D4 [×2], D4, Q8 [×2], Q8 [×5], C23, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×9], C4⋊C4 [×13], C2×C8 [×4], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C8⋊C4 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8 [×5], C2.D8 [×4], C2×C4⋊C4, C4×D4 [×3], C4×Q8 [×4], C4×Q8, C22⋊Q8 [×3], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C4×SD16, SD16⋊C4 [×2], C8⋊4Q8, Q8⋊Q8, D4⋊2Q8, D4.Q8 [×2], Q8.Q8 [×2], C8.5Q8, C8⋊Q8 [×2], D4⋊3Q8, Q8⋊3Q8, SD16⋊3Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C22×D4, C22×Q8, 2- (1+4), D4×Q8, D8⋊C22, D4○SD16, SD16⋊3Q8

Generators and relations
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, cbc-1=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 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 60)(10 63)(11 58)(12 61)(13 64)(14 59)(15 62)(16 57)(25 47)(26 42)(27 45)(28 48)(29 43)(30 46)(31 41)(32 44)(33 55)(34 50)(35 53)(36 56)(37 51)(38 54)(39 49)(40 52)

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
0000512
000055
0051200
005500
,
100000
010000
0016000
000100
0000160
000001
,
0160000
100000
000001
0000160
000100
0016000
,
0130000
1300000
0001600
001000
000001
0000160

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

Character table of SD16⋊3Q8

 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
ρ254-44-40040-400000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-4-440004i04i0000000000000000000    complex lifted from D8⋊C22
ρ274-4-440004i04i0000000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16
ρ2944-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

SD_{16}\rtimes_3Q_8
 
% in TeX
 
G:=Group("SD16:3Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2120);
 
// by ID
 
G=gap.SmallGroup(128,2120);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,346,304,4037,1027,124]);
 
// 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,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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