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G = SD16⋊2Q8  order 128 = 27

2nd semidirect product of SD16 and Q8 acting via Q8/C4=C2

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

Aliases: SD16⋊2Q8, C42.67C23, C4.1012- (1+4), C8⋊3(C2×Q8), C8⋊Q8⋊33C2, Q8⋊5(C2×Q8), C2.44(D4×Q8), C4⋊C4.389D4, C8⋊2Q8⋊34C2, C8⋊4Q8⋊16C2, (D4×Q8).10C2, D4.12(C2×Q8), D4.Q8.3C2, C4.Q16⋊47C2, Q8⋊Q8⋊28C2, Q8⋊3Q8⋊10C2, (C2×Q8).250D4, C2.68(Q8○D8), (C4×SD16).6C2, C4.44(C22×Q8), C4⋊C4.275C23, C4⋊C8.145C22, C4.80(C8⋊C22), (C4×C8).202C22, (C2×C8).215C23, (C2×C4).578C24, D4⋊Q8.13C2, C4⋊Q8.207C22, C4.Q8.78C22, C8⋊C4.71C22, SD16⋊C4.4C2, (C2×D4).439C23, (C4×D4).214C22, (C2×Q8).412C23, (C4×Q8).205C22, C2.D8.139C22, D4⋊C4.95C22, C22.838(C22×D4), C42.C2.76C22, Q8⋊C4.166C22, (C2×SD16).124C22, (C2×C4).648(C2×D4), C2.91(C2×C8⋊C22), SmallGroup(128,2118)

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

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

Subgroups: 320 in 180 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 [×13], D4 [×2], D4, Q8 [×2], Q8 [×11], 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 [×8], 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], C4×D4, C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×3], C42.C2 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C22×Q8, C4×SD16, SD16⋊C4 [×2], C8⋊4Q8, D4⋊Q8, Q8⋊Q8, C4.Q16 [×2], D4.Q8 [×2], C8⋊2Q8, C8⋊Q8 [×2], D4×Q8, Q8⋊3Q8, SD16⋊2Q8

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, Q8○D8, SD16⋊2Q8

Generators and relations
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, 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 57)(10 60)(11 63)(12 58)(13 61)(14 64)(15 59)(16 62)(25 53)(26 56)(27 51)(28 54)(29 49)(30 52)(31 55)(32 50)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
0016877
00916107
0013419
00131381
,
100000
010000
0016000
000100
001010
00016016
,
1150000
1160000
0016000
0001600
0000160
0000016
,
1070000
570000
00016015
001020
000101
00160160

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,9,13,13,0,0,8,16,4,13,0,0,7,10,1,8,0,0,7,7,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,1,0,16,0,0,16,0,1,0,0,0,0,2,0,16,0,0,15,0,1,0] >;
 

Character table of SD16⋊2Q8

 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
ρ254-4-4400040-40000000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44000-4040000000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40040-400000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2844-4-40000000000000000000002√22√200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-40000000000000000000002√22√200    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

SD_{16}\rtimes_2Q_8
 
% in TeX
 
G:=Group("SD16:2Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2118);
 
// by ID
 
G=gap.SmallGroup(128,2118);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,723,346,80,4037,1027,124]);
 
// 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,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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