Copied to
clipboard

G = Q8⋊D8order 128 = 27

1st semidirect product of Q8 and D8 acting via D8/D4=C2

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

Aliases: Q82D8, C42.182C23, Q8⋊C87C2, D4⋊C810C2, C4⋊D82C2, C84D42C2, C4⋊C4.47D4, C4.25(C2×D8), Q86D41C2, (C2×D4).43D4, C4.D86C2, C4⋊SD1630C2, C4.80(C4○D8), (C4×C8).41C22, (C2×Q8).193D4, C4⋊C8.158C22, C4.56(C8⋊C22), (C4×D4).18C22, C41D4.6C22, (C4×Q8).18C22, C2.11(C22⋊D8), C2.14(D44D4), C2.13(D4⋊D4), C22.148C22≀C2, (C2×C4).939(C2×D4), SmallGroup(128,353)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — Q8⋊D8
C1C2C22C2×C4C42C4×Q8Q86D4 — Q8⋊D8
C1C22C42 — Q8⋊D8
C1C22C42 — Q8⋊D8
C1C22C22C42 — Q8⋊D8

Generators and relations for Q8⋊D8
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 400 in 139 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×5], C22, C22 [×12], C8 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×20], Q8 [×2], Q8, C23 [×4], C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4, C2×C8 [×3], D8 [×6], SD16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C4×C8, D4⋊C4 [×2], C4⋊C8 [×2], C4×D4, C4×D4, C4×Q8, C4⋊D4 [×3], C41D4 [×2], C41D4, C2×D8 [×3], C2×SD16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, C4⋊D8, C4⋊SD16, C84D4, Q86D4, Q8⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C4○D8, C8⋊C22 [×2], C22⋊D8, D4⋊D4, D44D4, Q8⋊D8

Character table of Q8⋊D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 111188816222244444844448888
ρ111111111111111111111111111    trivial
ρ211111-1-111111-111-11-1-1-1-1-1-11-11    linear of order 2
ρ31111-111-11111-1-11-1-1-1-1-1-1-11111    linear of order 2
ρ41111-1-1-1-111111-111-111111-11-11    linear of order 2
ρ51111111-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ611111-1-1-11111-111-11-111111-11-1    linear of order 2
ρ71111-11111111-1-11-1-1-11111-1-1-1-1    linear of order 2
ρ81111-1-1-1111111-111-11-1-1-1-11-11-1    linear of order 2
ρ922220000-2-22220-220-200000000    orthogonal lifted from D4
ρ1022220000-2-222-20-2-20200000000    orthogonal lifted from D4
ρ1122220-220-2-2-2-200200000000000    orthogonal lifted from D4
ρ122222200022-2-20-2-20-2000000000    orthogonal lifted from D4
ρ132222-200022-2-202-202000000000    orthogonal lifted from D4
ρ14222202-20-2-2-2-200200000000000    orthogonal lifted from D4
ρ152-22-20000002-2200-2002-22-20-202    orthogonal lifted from D8
ρ162-22-20000002-2200-200-22-22020-2    orthogonal lifted from D8
ρ172-22-20000002-2-2002002-22-2020-2    orthogonal lifted from D8
ρ182-22-20000002-2-200200-22-220-202    orthogonal lifted from D8
ρ1922-2-200002-20002i00-2i0-2-222-20--20    complex lifted from C4○D8
ρ2022-2-200002-2000-2i002i0-2-222--20-20    complex lifted from C4○D8
ρ2122-2-200002-2000-2i002i022-2-2-20--20    complex lifted from C4○D8
ρ2222-2-200002-20002i00-2i022-2-2--20-20    complex lifted from C4○D8
ρ234-4-44000000000000002-2-220000    orthogonal lifted from D44D4
ρ244-44-4000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ254-4-4400000000000000-222-20000    orthogonal lifted from D44D4
ρ2644-4-40000-440000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of Q8⋊D8
On 64 points
Generators in S64
(1 63 27 46)(2 64 28 47)(3 57 29 48)(4 58 30 41)(5 59 31 42)(6 60 32 43)(7 61 25 44)(8 62 26 45)(9 52 33 24)(10 53 34 17)(11 54 35 18)(12 55 36 19)(13 56 37 20)(14 49 38 21)(15 50 39 22)(16 51 40 23)
(1 10 27 34)(2 54 28 18)(3 36 29 12)(4 20 30 56)(5 14 31 38)(6 50 32 22)(7 40 25 16)(8 24 26 52)(9 62 33 45)(11 47 35 64)(13 58 37 41)(15 43 39 60)(17 46 53 63)(19 57 55 48)(21 42 49 59)(23 61 51 44)
(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)
(2 8)(3 7)(4 6)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 29)(26 28)(30 32)(41 60)(42 59)(43 58)(44 57)(45 64)(46 63)(47 62)(48 61)

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

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

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

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

0100
16000
0010
0001
,
121200
12500
0010
0001
,
31400
3300
00011
0036
,
1000
01600
0010
001616
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,0,3,0,0,11,6],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

Q8⋊D8 in GAP, Magma, Sage, TeX

Q_8\rtimes D_8
% in TeX

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

G:=SmallGroup(128,353);
// by ID

G=gap.SmallGroup(128,353);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of Q8⋊D8 in TeX

׿
×
𝔽