Copied to
clipboard

G = D8⋊Q8order 128 = 27

2nd semidirect product of D8 and Q8 acting via Q8/C4=C2

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

Aliases: D82Q8, C4.10SD32, C42.147D4, C4⋊C1612C2, (C4×D8).7C2, C164C47C2, (C2×C8).71D4, C8.31(C2×Q8), C82Q810C2, (C2×C4).153D8, C2.D16.5C2, C8.65(C4○D4), (C4×C8).67C22, C2.11(C2×SD32), (C2×C16).43C22, (C2×C8).528C23, C22.114(C2×D8), C4.47(C22⋊Q8), C2.14(C16⋊C22), C2.D8.13C22, C2.14(D4⋊Q8), (C2×D8).113C22, C4.10(C8.C22), (C2×C4).796(C2×D4), SmallGroup(128,958)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8⋊Q8
C1C2C4C8C2×C8C2×D8C4×D8 — D8⋊Q8
C1C2C4C2×C8 — D8⋊Q8
C1C22C42C4×C8 — D8⋊Q8
C1C2C2C2C2C4C4C2×C8 — D8⋊Q8

Generators and relations for D8⋊Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=dad-1=a-1, ac=ca, cbc-1=a4b, 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, C164C4 [×2], C4×D8, C82Q8, D8⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, SD32 [×2], C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C2×SD32, C16⋊C22, D8⋊Q8

Character table of D8⋊Q8

 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
ρ14222200-222-2-200000000002-22-2-222-2    orthogonal lifted from D8
ρ152-2-222-2200-200000-222-20000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22-22200-200000-222-20000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-2200200-20-2i2i002-2-220000000000    complex lifted from C4○D4
ρ182-2-2200200-202i-2i002-2-220000000000    complex lifted from C4○D4
ρ192-22-2000-220000002-22-2-22ζ1615169ζ165163ζ16716ζ165163ζ16131611ζ1615169ζ16716ζ16131611    complex lifted from SD32
ρ202-22-2000-220000002-22-2-22ζ16716ζ16131611ζ1615169ζ16131611ζ165163ζ16716ζ1615169ζ165163    complex lifted from SD32
ρ212-22-2000-22000000-22-222-2ζ16131611ζ1615169ζ165163ζ1615169ζ16716ζ16131611ζ165163ζ16716    complex lifted from SD32
ρ222-22-2000-22000000-22-222-2ζ165163ζ16716ζ16131611ζ16716ζ1615169ζ165163ζ16131611ζ1615169    complex lifted from SD32
ρ232-22-20002-20000002-22-22-2ζ16716ζ16131611ζ1615169ζ165163ζ165163ζ1615169ζ16716ζ16131611    complex lifted from SD32
ρ242-22-20002-20000002-22-22-2ζ1615169ζ165163ζ16716ζ16131611ζ16131611ζ16716ζ1615169ζ165163    complex lifted from SD32
ρ252-22-20002-2000000-22-22-22ζ16131611ζ1615169ζ165163ζ16716ζ16716ζ165163ζ16131611ζ1615169    complex lifted from SD32
ρ262-22-20002-2000000-22-22-22ζ165163ζ16716ζ16131611ζ1615169ζ1615169ζ16131611ζ165163ζ16716    complex lifted from SD32
ρ2744-4-400000000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ2844-4-4000000000002222-22-220000000000    orthogonal lifted from C16⋊C22
ρ294-4-4400-40040000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D8⋊Q8
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 25 15 21)(2 26 16 22)(3 27 9 23)(4 28 10 24)(5 29 11 17)(6 30 12 18)(7 31 13 19)(8 32 14 20)(33 55 45 59)(34 56 46 60)(35 49 47 61)(36 50 48 62)(37 51 41 63)(38 52 42 64)(39 53 43 57)(40 54 44 58)
(1 41 15 37)(2 48 16 36)(3 47 9 35)(4 46 10 34)(5 45 11 33)(6 44 12 40)(7 43 13 39)(8 42 14 38)(17 59 29 55)(18 58 30 54)(19 57 31 53)(20 64 32 52)(21 63 25 51)(22 62 26 50)(23 61 27 49)(24 60 28 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), (1,25,15,21)(2,26,16,22)(3,27,9,23)(4,28,10,24)(5,29,11,17)(6,30,12,18)(7,31,13,19)(8,32,14,20)(33,55,45,59)(34,56,46,60)(35,49,47,61)(36,50,48,62)(37,51,41,63)(38,52,42,64)(39,53,43,57)(40,54,44,58), (1,41,15,37)(2,48,16,36)(3,47,9,35)(4,46,10,34)(5,45,11,33)(6,44,12,40)(7,43,13,39)(8,42,14,38)(17,59,29,55)(18,58,30,54)(19,57,31,53)(20,64,32,52)(21,63,25,51)(22,62,26,50)(23,61,27,49)(24,60,28,56)>;

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

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

Matrix representation of D8⋊Q8 in GL4(𝔽17) generated by

31400
3300
0010
0001
,
31400
141400
0010
0001
,
0100
16000
0012
001616
,
101600
16700
001010
00127
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,1,16,0,0,2,16],[10,16,0,0,16,7,0,0,0,0,10,12,0,0,10,7] >;

D8⋊Q8 in GAP, Magma, Sage, TeX

D_8\rtimes Q_8
% in TeX

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

G:=SmallGroup(128,958);
// by ID

G=gap.SmallGroup(128,958);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,504,141,64,422,1684,438,242,4037,1027,124]);
// 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,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D8⋊Q8 in TeX

׿
×
𝔽