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G = D84Q8order 128 = 27

4th semidirect product of D8 and Q8 acting via Q8/C4=C2

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

Aliases: D84Q8, C42.65C23, C4.992- 1+4, C82(C2×Q8), C8⋊Q831C2, D45(C2×Q8), (D4×Q8)⋊14C2, C2.42(D4×Q8), C4⋊C4.387D4, C84Q814C2, C83Q811C2, (C4×D8).17C2, D4.Q851C2, D4⋊Q846C2, D43Q816C2, D42Q828C2, D8⋊C4.2C2, (C2×Q8).248D4, C4.42(C22×Q8), C4⋊C4.273C23, C4⋊C8.143C22, C4.52(C8⋊C22), (C2×C8).375C23, (C2×C4).576C24, (C4×C8).200C22, C4⋊Q8.205C22, C8⋊C4.69C22, (C2×D8).168C22, (C4×D4).212C22, (C2×D4).437C23, (C4×Q8).203C22, C4.Q8.118C22, C2.D8.138C22, C2.108(D4○SD16), D4⋊C4.93C22, C22.836(C22×D4), C42.C2.74C22, (C2×C4).646(C2×D4), C2.90(C2×C8⋊C22), SmallGroup(128,2116)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D84Q8
C1C2C4C2×C4C42C4×D4D43Q8 — D84Q8
C1C2C2×C4 — D84Q8
C1C22C4×Q8 — D84Q8
C1C2C2C2×C4 — D84Q8

Generators and relations for D84Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 360 in 187 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×8], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×16], D4 [×4], D4 [×2], Q8 [×9], C23 [×2], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×6], C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4×C8, C8⋊C4 [×2], D4⋊C4 [×2], D4⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×6], C2.D8, C2.D8 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4 [×4], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×D8, C22×Q8, C4×D8, D8⋊C4 [×2], C84Q8, D4⋊Q8, D42Q8, D42Q8 [×2], D4.Q8 [×2], C83Q8, C8⋊Q8 [×2], D4×Q8, D43Q8, D84Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C8⋊C22 [×2], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8⋊C22, D4○SD16, D84Q8

Character table of D84Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114444222244444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-1-11-1-1-111-11-11111-1-11-1    linear of order 2
ρ311111-11-111-1-11-1-1-11111-1-1-1-1-111-11    linear of order 2
ρ41111-11-111111111111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1111111111-1-1-1-1-1-1111111    linear of order 2
ρ61111111111-1-11-1-1-11-11-11-1-111-1-11-1    linear of order 2
ρ71111-11-1111-1-11-1-1-11-1-1-1111-1-111-11    linear of order 2
ρ811111-11-1111111111-11-1-111-1-1-1-1-1-1    linear of order 2
ρ91111-11-111111-1-11-1-111-1-11-1-1-1-1-111    linear of order 2
ρ1011111-11-111-1-1-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ111111-1-1-1-111-1-1-11-11-111-11-1111-1-1-11    linear of order 2
ρ12111111111111-1-11-1-11-1-1-1-111111-1-1    linear of order 2
ρ1311111-11-11111-1-11-1-1-1-111-11-1-1-1-111    linear of order 2
ρ141111-11-1111-1-1-11-11-1-111-1-11-1-1111-1    linear of order 2
ρ151111111111-1-1-11-11-1-1-11-11-111-1-1-11    linear of order 2
ρ161111-1-1-1-11111-1-11-1-1-11111-11111-1-1    linear of order 2
ρ1722220000-2-2-2-2222-2-2000000000000    orthogonal lifted from D4
ρ1822220000-2-222-22-2-22000000000000    orthogonal lifted from D4
ρ1922220000-2-2-2-2-2-2222000000000000    orthogonal lifted from D4
ρ2022220000-2-2222-2-22-2000000000000    orthogonal lifted from D4
ρ212-22-2-2-222-2200000000000002-20000    symplectic lifted from Q8, Schur index 2
ρ222-22-222-2-2-2200000000000002-20000    symplectic lifted from Q8, Schur index 2
ρ232-22-22-2-22-220000000000000-220000    symplectic lifted from Q8, Schur index 2
ρ242-22-2-222-2-220000000000000-220000    symplectic lifted from Q8, Schur index 2
ρ254-4-440000004-400000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44000000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-400004-40000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of D84Q8
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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 36)(10 35)(11 34)(12 33)(13 40)(14 39)(15 38)(16 37)(17 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
(1 34 29 11)(2 35 30 12)(3 36 31 13)(4 37 32 14)(5 38 25 15)(6 39 26 16)(7 40 27 9)(8 33 28 10)(17 47 55 64)(18 48 56 57)(19 41 49 58)(20 42 50 59)(21 43 51 60)(22 44 52 61)(23 45 53 62)(24 46 54 63)
(1 57 29 48)(2 62 30 45)(3 59 31 42)(4 64 32 47)(5 61 25 44)(6 58 26 41)(7 63 27 46)(8 60 28 43)(9 24 40 54)(10 21 33 51)(11 18 34 56)(12 23 35 53)(13 20 36 50)(14 17 37 55)(15 22 38 52)(16 19 39 49)

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

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

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

Matrix representation of D84Q8 in GL6(𝔽17)

100000
010000
0000125
00001212
0051200
005500
,
100000
010000
0016000
000100
0000160
000001
,
120000
16160000
000010
000001
0016000
0001600
,
8140000
1690000
0070160
0007016
00160100
00016010

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,12,12,0,0,0,0,5,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[8,16,0,0,0,0,14,9,0,0,0,0,0,0,7,0,16,0,0,0,0,7,0,16,0,0,16,0,10,0,0,0,0,16,0,10] >;

D84Q8 in GAP, Magma, Sage, TeX

D_8\rtimes_4Q_8
% in TeX

G:=Group("D8:4Q8");
// GroupNames label

G:=SmallGroup(128,2116);
// by ID

G=gap.SmallGroup(128,2116);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of D84Q8 in TeX

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