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G = SD163Q8order 128 = 27

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

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

Aliases: SD163Q8, C42.69C23, C4.1032- 1+4, C8⋊Q834C2, C8.8(C2×Q8), C2.46(D4×Q8), C4⋊C4.391D4, C84Q817C2, Q8.Q852C2, D4.13(C2×Q8), Q8.13(C2×Q8), D4.Q8.4C2, Q8⋊Q829C2, Q83Q811C2, D42Q8.3C2, (C2×Q8).140D4, C8.5Q824C2, (C4×SD16).7C2, C4.46(C22×Q8), C4⋊C4.277C23, C4⋊C8.147C22, (C2×C4).580C24, (C2×C8).377C23, (C4×C8).204C22, D43Q8.11C2, C4⋊Q8.209C22, C8⋊C4.73C22, SD16⋊C4.5C2, (C4×D4).215C22, (C2×D4).440C23, (C4×Q8).207C22, (C2×Q8).414C23, 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

C1C2×C4 — SD163Q8
C1C2C4C2×C4C42C4×D4D43Q8 — SD163Q8
C1C2C2×C4 — SD163Q8
C1C22C4×Q8 — SD163Q8
C1C2C2C2×C4 — SD163Q8

Generators and relations for SD163Q8
 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 >

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], C84Q8, Q8⋊Q8, D42Q8, D4.Q8 [×2], Q8.Q8 [×2], C8.5Q8, C8⋊Q8 [×2], D43Q8, Q83Q8, SD163Q8
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, SD163Q8

Character table of SD163Q8

 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
ρ264-4-44000-4i04i0000000000000000000    complex lifted from D8⋊C22
ρ274-4-440004i0-4i0000000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of SD163Q8
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 52)(10 55)(11 50)(12 53)(13 56)(14 51)(15 54)(16 49)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)(33 61)(34 64)(35 59)(36 62)(37 57)(38 60)(39 63)(40 58)
(1 16 23 53)(2 9 24 54)(3 10 17 55)(4 11 18 56)(5 12 19 49)(6 13 20 50)(7 14 21 51)(8 15 22 52)(25 64 41 36)(26 57 42 37)(27 58 43 38)(28 59 44 39)(29 60 45 40)(30 61 46 33)(31 62 47 34)(32 63 48 35)
(1 26 23 42)(2 31 24 47)(3 28 17 44)(4 25 18 41)(5 30 19 46)(6 27 20 43)(7 32 21 48)(8 29 22 45)(9 34 54 62)(10 39 55 59)(11 36 56 64)(12 33 49 61)(13 38 50 58)(14 35 51 63)(15 40 52 60)(16 37 53 57)

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

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

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

Matrix representation of SD163Q8 in 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] >;

SD163Q8 in GAP, Magma, Sage, TeX

{\rm 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

Export

Character table of SD163Q8 in TeX

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