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G = Q169D4order 128 = 27

3rd semidirect product of Q16 and D4 acting via D4/C22=C2

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

Aliases: Q169D4, C42.37C23, C4.1242+ 1+4, C2.54D42, (D4×Q8)⋊3C2, C89D47C2, C8.33(C2×D4), C88D419C2, C4⋊C4.149D4, Q8.20(C2×D4), Q8⋊D416C2, C42Q1635C2, (C2×D4).308D4, C8.D420C2, C8.2D419C2, C4⋊C8.93C22, (C2×C8).87C23, Q85D4.1C2, C2.38(Q8○D8), C22⋊C4.41D4, C4.84(C22×D4), Q16⋊C419C2, D4.7D436C2, C4⋊C4.209C23, (C2×C4).468C24, Q8.D436C2, C22⋊Q1627C2, (C22×Q16)⋊20C2, C23.462(C2×D4), C4⋊Q8.134C22, C8⋊C4.37C22, C4.Q8.52C22, (C4×D4).144C22, D4⋊C4.9C22, (C2×D4).208C23, C4⋊D4.59C22, C22⋊C8.71C22, C223(C8.C22), (C2×Q16).81C22, (C2×Q8).387C23, (C4×Q8).138C22, C22⋊Q8.58C22, (C22×C4).319C23, (C22×C8).284C22, Q8⋊C4.66C22, (C2×SD16).48C22, C4.4D4.53C22, C22.728(C22×D4), (C22×Q8).329C22, (C2×M4(2)).103C22, (C2×C4).592(C2×D4), (C2×C8.C22)⋊28C2, C2.71(C2×C8.C22), (C2×C4○D4).185C22, SmallGroup(128,2002)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q169D4
C1C2C22C2×C4C2×Q8C22×Q8C22×Q16 — Q169D4
C1C2C2×C4 — Q169D4
C1C22C4×D4 — Q169D4
C1C2C2C2×C4 — Q169D4

Generators and relations for Q169D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, cac-1=dad=a3, bc=cb, bd=db, dcd=c-1 >

Subgroups: 448 in 236 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×8], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×19], D4 [×8], Q8 [×4], Q8 [×14], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×5], Q16 [×4], Q16 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×5], C2×Q8 [×12], C4○D4 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4, Q8⋊C4 [×5], C4⋊C8, C4.Q8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×3], C2×Q16 [×6], C2×Q16 [×4], C8.C22 [×4], C22×Q8 [×3], C2×C4○D4, C89D4, Q16⋊C4, Q8⋊D4, C22⋊Q16 [×2], D4.7D4, C42Q16, Q8.D4, C88D4, C8.D4, C8.2D4, Q85D4, D4×Q8, C22×Q16, C2×C8.C22, Q169D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8.C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8.C22, Q8○D8, Q169D4

Character table of Q169D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11112248224444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-11111-1111-111-11-1-111-1-1-11    linear of order 2
ρ31111111-1111-111-1111-1111-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-11111-1-11-11-11-1-11-11-1-1111-1    linear of order 2
ρ51111-1-1111111-111-11-1-11-1-1-111-1-11-1    linear of order 2
ρ6111111-1-11111111-1-1-1-1-1-1111111-1-1    linear of order 2
ρ71111-1-11-1111-1-11-1-11-111-1-11-1-111-11    linear of order 2
ρ8111111-11111-111-1-1-1-11-1-11-1-1-1-1-111    linear of order 2
ρ91111-1-1-1-111-11-1-111-1111-11-1-1-1111-1    linear of order 2
ρ101111111111-111-111111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ111111-1-1-1111-1-1-1-1-11-11-11-11111-1-1-11    linear of order 2
ρ121111111-111-1-11-1-1111-1-1-1-1-1111111    linear of order 2
ρ13111111-1-111-111-11-1-1-1-111-11-1-1-1-111    linear of order 2
ρ141111-1-11111-11-1-11-11-1-1-111-1-1-111-11    linear of order 2
ρ15111111-1111-1-11-1-1-1-1-1111-1-11111-1-1    linear of order 2
ρ161111-1-11-111-1-1-1-1-1-11-11-111111-1-11-1    linear of order 2
ρ172-22-200002-2-2202-200000000-220000    orthogonal lifted from D4
ρ182222-2-2-20-2-20020022-200000000000    orthogonal lifted from D4
ρ192222-2-220-2-200200-2-2200000000000    orthogonal lifted from D4
ρ202-22-200002-2220-2-2000000002-20000    orthogonal lifted from D4
ρ21222222-20-2-200-200-22200000000000    orthogonal lifted from D4
ρ2222222220-2-200-2002-2-200000000000    orthogonal lifted from D4
ρ232-22-200002-22-20-2200000000-220000    orthogonal lifted from D4
ρ242-22-200002-2-2-2022000000002-20000    orthogonal lifted from D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of Q169D4
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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 34 19 48)(2 37 20 43)(3 40 21 46)(4 35 22 41)(5 38 23 44)(6 33 24 47)(7 36 17 42)(8 39 18 45)(9 56 58 31)(10 51 59 26)(11 54 60 29)(12 49 61 32)(13 52 62 27)(14 55 63 30)(15 50 64 25)(16 53 57 28)
(2 4)(3 7)(6 8)(9 64)(10 59)(11 62)(12 57)(13 60)(14 63)(15 58)(16 61)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)(49 53)(50 56)(52 54)

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

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

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

Matrix representation of Q169D4 in GL6(𝔽17)

1600000
0160000
001414413
003141313
0044314
0041333
,
1600000
0160000
000010
000001
0016000
0001600
,
0160000
100000
000107
0010100
0001001
007010
,
100000
0160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,14,3,4,4,0,0,14,14,4,13,0,0,4,13,3,3,0,0,13,13,14,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,7,0,0,1,0,10,0,0,0,0,10,0,1,0,0,7,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

Q169D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_9D_4
% in TeX

G:=Group("Q16:9D4");
// GroupNames label

G:=SmallGroup(128,2002);
// by ID

G=gap.SmallGroup(128,2002);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,346,248,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q169D4 in TeX

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