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G = Q83D8order 128 = 27

2nd semidirect product of Q8 and D8 acting via D8/D4=C2

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

Aliases: Q83D8, D46SD16, C42.188C23, Q8⋊C88C2, (D4×Q8)⋊1C2, D4⋊C816C2, C4⋊C4.24D4, C4.27(C2×D8), C4⋊D8.2C2, C4.4D82C2, C4⋊SD1632C2, (C2×D4).250D4, C4.10D88C2, (C4×C8).16C22, (C2×Q8).195D4, C4.26(C2×SD16), C4⋊Q8.10C22, C4⋊C8.161C22, C4.59(C8⋊C22), (C4×D4).22C22, (C4×Q8).22C22, C2.13(C22⋊D8), C2.13(Q8⋊D4), C41D4.12C22, C4.58(C8.C22), C2.14(D4.8D4), C22.154C22≀C2, (C2×C4).945(C2×D4), SmallGroup(128,359)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — Q83D8
C1C2C22C2×C4C42C4×Q8D4×Q8 — Q83D8
C1C22C42 — Q83D8
C1C22C42 — Q83D8
C1C22C22C42 — Q83D8

Generators and relations for Q83D8
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 320 in 125 conjugacy classes, 38 normal (32 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×7], C22, C22 [×7], C8 [×3], C2×C4 [×3], C2×C4 [×11], D4 [×2], D4 [×7], Q8 [×2], Q8 [×7], C23 [×2], C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×3], D8 [×2], SD16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×7], C4×C8, D4⋊C4 [×4], C4⋊C8 [×2], C4×D4, C4×D4, C4×Q8, C22⋊Q8 [×3], C41D4, C4⋊Q8, C4⋊Q8, C2×D8, C2×SD16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊D8, C4⋊SD16, C4.4D8, D4×Q8, Q83D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], SD16 [×2], C2×D4 [×3], C22≀C2, C2×D8, C2×SD16, C8⋊C22, C8.C22, C22⋊D8, Q8⋊D4, D4.8D4, Q83D8

Character table of Q83D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111144162222444888844448888
ρ111111111111111111111111111    trivial
ρ2111111111111-1-1-11-1-1-1-1-1-1-111-1    linear of order 2
ρ31111-1-1111111111-1-1-1-1-1-1-11-1-11    linear of order 2
ρ41111-1-1111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ5111111-111111-1-1-11-1-111111-1-11    linear of order 2
ρ6111111-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-111111-1-1-1-111-1-1-1-11111    linear of order 2
ρ81111-1-1-111111111-1-1-11111-111-1    linear of order 2
ρ92222000-2-2-2-2200002-200000000    orthogonal lifted from D4
ρ102222000-2-2-2-220000-2200000000    orthogonal lifted from D4
ρ112222000-2-222-2-2-2200000000000    orthogonal lifted from D4
ρ122222000-2-222-222-200000000000    orthogonal lifted from D4
ρ132222-2-2022-2-2-200020000000000    orthogonal lifted from D4
ρ14222222022-2-2-2000-20000000000    orthogonal lifted from D4
ρ1522-2-200000-220-22000022-2-20-220    orthogonal lifted from D8
ρ1622-2-200000-220-220000-2-22202-20    orthogonal lifted from D8
ρ1722-2-200000-2202-2000022-2-202-20    orthogonal lifted from D8
ρ1822-2-200000-2202-20000-2-2220-220    orthogonal lifted from D8
ρ192-2-222-202-2000000000--2-2-2--2--200-2    complex lifted from SD16
ρ202-2-22-2202-2000000000-2--2--2-2--200-2    complex lifted from SD16
ρ212-2-222-202-2000000000-2--2--2-2-200--2    complex lifted from SD16
ρ222-2-22-2202-2000000000--2-2-2--2-200--2    complex lifted from SD16
ρ2344-4-4000004-4000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44000-4400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-400000000000000-2i2i-2i2i0000    complex lifted from D4.8D4
ρ264-44-4000000000000002i-2i2i-2i0000    complex lifted from D4.8D4

Smallest permutation representation of Q83D8
On 64 points
Generators in S64
(1 10 54 32)(2 25 55 11)(3 12 56 26)(4 27 49 13)(5 14 50 28)(6 29 51 15)(7 16 52 30)(8 31 53 9)(17 33 42 58)(18 59 43 34)(19 35 44 60)(20 61 45 36)(21 37 46 62)(22 63 47 38)(23 39 48 64)(24 57 41 40)
(1 43 54 18)(2 35 55 60)(3 45 56 20)(4 37 49 62)(5 47 50 22)(6 39 51 64)(7 41 52 24)(8 33 53 58)(9 42 31 17)(10 59 32 34)(11 44 25 19)(12 61 26 36)(13 46 27 21)(14 63 28 38)(15 48 29 23)(16 57 30 40)
(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 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 60)(18 59)(19 58)(20 57)(21 64)(22 63)(23 62)(24 61)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)(49 51)(52 56)(53 55)

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

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

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

Matrix representation of Q83D8 in GL4(𝔽17) generated by

01600
1000
0010
0001
,
121200
12500
0010
0001
,
0100
1000
00143
001414
,
0100
1000
0010
00016
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,14,14,0,0,3,14],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,16] >;

Q83D8 in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_8
% in TeX

G:=Group("Q8:3D8");
// GroupNames label

G:=SmallGroup(128,359);
// by ID

G=gap.SmallGroup(128,359);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of Q83D8 in TeX

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