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G = D81Q8order 128 = 27

1st semidirect product of D8 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D81Q8, C4.13D16, C42.145D4, C4⋊C166C2, C82Q89C2, (C4×D8).6C2, C163C44C2, C2.7(C2×D16), (C2×C8).69D4, C8.29(C2×Q8), (C2×C4).151D8, C2.D16.3C2, C8.63(C4○D4), (C4×C8).65C22, (C2×C16).7C22, (C2×C8).526C23, C22.112(C2×D8), C4.8(C8.C22), C4.45(C22⋊Q8), C2.D8.11C22, C2.12(D4⋊Q8), C2.14(Q32⋊C2), (C2×D8).112C22, (C2×C4).794(C2×D4), SmallGroup(128,956)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D81Q8
C1C2C4C8C2×C8C2×D8C4×D8 — D81Q8
C1C2C4C2×C8 — D81Q8
C1C22C42C4×C8 — D81Q8
C1C2C2C2C2C4C4C2×C8 — D81Q8

Generators and relations for D81Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 196 in 71 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×4], C2×C8 [×2], D8 [×2], D8, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, C2.D8, C2.D8 [×2], C2.D8, C2×C16 [×2], C4×D4, C4⋊Q8, C2×D8, C2.D16 [×2], C4⋊C16, C163C4 [×2], C4×D8, C82Q8, D81Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, D16 [×2], C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C2×D16, Q32⋊C2, D81Q8

Character table of D81Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111882222488161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111111-1-1-1-111111111111111    linear of order 2
ρ41111-1-111111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-11-1-11-1111-11111-1-1111-11-1-1-1    linear of order 2
ρ61111-1-11-1-11-111-111111-1-1-1-1-11-1111    linear of order 2
ρ71111111-1-11-1-1-1-111111-1-1111-11-1-1-1    linear of order 2
ρ81111111-1-11-1-1-11-11111-1-1-1-1-11-1111    linear of order 2
ρ92222002-2-22-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ10222200-222-2-20000000000-22-222-2-22    orthogonal lifted from D8
ρ11222200222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ12222200-2-2-2-220000000000-22-2-2222-2    orthogonal lifted from D8
ρ13222200-2-2-2-2200000000002-222-2-2-22    orthogonal lifted from D8
ρ142-22-2000-220000002-22-2-22ζ1651631671616516316716ζ16716ζ165163165163ζ16716    orthogonal lifted from D16
ρ15222200-222-2-200000000002-22-2-222-2    orthogonal lifted from D8
ρ162-22-20002-20000002-22-22-2ζ16516316716165163ζ16716ζ16716165163ζ16516316716    orthogonal lifted from D16
ρ172-22-2000-22000000-22-222-216716165163ζ16716165163ζ16516316716ζ16716ζ165163    orthogonal lifted from D16
ρ182-22-2000-22000000-22-222-2ζ16716ζ16516316716ζ165163165163ζ1671616716165163    orthogonal lifted from D16
ρ192-22-20002-2000000-22-22-2216716165163ζ16716ζ165163ζ165163ζ1671616716165163    orthogonal lifted from D16
ρ202-22-2000-220000002-22-2-22165163ζ16716ζ165163ζ1671616716165163ζ16516316716    orthogonal lifted from D16
ρ212-22-20002-20000002-22-22-2165163ζ16716ζ1651631671616716ζ165163165163ζ16716    orthogonal lifted from D16
ρ222-22-20002-2000000-22-22-22ζ16716ζ1651631671616516316516316716ζ16716ζ165163    orthogonal lifted from D16
ρ232-2-222-2200-200000-222-20000000000    symplectic lifted from Q8, Schur index 2
ρ242-2-22-22200-200000-222-20000000000    symplectic lifted from Q8, Schur index 2
ρ252-2-2200200-20-2i2i002-2-220000000000    complex lifted from C4○D4
ρ262-2-2200200-202i-2i002-2-220000000000    complex lifted from C4○D4
ρ274-4-4400-40040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-4000000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of D81Q8
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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)(49 55)(50 54)(51 53)(57 63)(58 62)(59 61)
(1 27 11 19)(2 28 12 20)(3 29 13 21)(4 30 14 22)(5 31 15 23)(6 32 16 24)(7 25 9 17)(8 26 10 18)(33 49 41 57)(34 50 42 58)(35 51 43 59)(36 52 44 60)(37 53 45 61)(38 54 46 62)(39 55 47 63)(40 56 48 64)
(1 45 11 37)(2 44 12 36)(3 43 13 35)(4 42 14 34)(5 41 15 33)(6 48 16 40)(7 47 9 39)(8 46 10 38)(17 63 25 55)(18 62 26 54)(19 61 27 53)(20 60 28 52)(21 59 29 51)(22 58 30 50)(23 57 31 49)(24 64 32 56)

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

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

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

Matrix representation of D81Q8 in GL4(𝔽17) generated by

1000
0100
00143
001414
,
1000
0100
00143
0033
,
0100
16000
0010
0001
,
7100
11000
0046
00613
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,14,14,0,0,3,14],[1,0,0,0,0,1,0,0,0,0,14,3,0,0,3,3],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[7,1,0,0,1,10,0,0,0,0,4,6,0,0,6,13] >;

D81Q8 in GAP, Magma, Sage, TeX

D_8\rtimes_1Q_8
% in TeX

G:=Group("D8:1Q8");
// GroupNames label

G:=SmallGroup(128,956);
// by ID

G=gap.SmallGroup(128,956);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D81Q8 in TeX

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