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G = D85Q8order 128 = 27

5th semidirect product of D8 and Q8 acting via Q8/C4=C2

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

Aliases: D85Q8, C42.70C23, C4.1042- 1+4, C8⋊Q835C2, C8.9(C2×Q8), C2.47(D4×Q8), C4⋊C4.392D4, C84Q818C2, (C4×D8).18C2, D4.Q852C2, D4.14(C2×Q8), D43Q817C2, D4⋊Q847C2, D8⋊C4.3C2, C2.69(D4○D8), (C2×Q8).141D4, C8.5Q825C2, C4.47(C22×Q8), C4⋊C8.148C22, C4⋊C4.278C23, (C2×C8).216C23, (C4×C8).205C22, (C2×C4).581C24, C4⋊Q8.210C22, C8⋊C4.74C22, C4.Q8.79C22, C2.D8.76C22, (C2×D4).441C23, (C4×D4).216C22, (C2×D8).169C22, (C4×Q8).208C22, D4⋊C4.97C22, C22.841(C22×D4), C42.C2.79C22, C2.106(D8⋊C22), (C2×C4).651(C2×D4), SmallGroup(128,2121)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D85Q8
C1C2C4C2×C4C42C4×D4D43Q8 — D85Q8
C1C2C2×C4 — D85Q8
C1C22C4×Q8 — D85Q8
C1C2C2C2×C4 — D85Q8

Generators and relations for D85Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, bd=db, dcd-1=c-1 >

Subgroups: 328 in 174 conjugacy classes, 94 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×8], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], D4 [×2], Q8 [×3], C23 [×2], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×6], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4 [×2], D4⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×3], C2.D8 [×2], C2×C4⋊C4 [×2], C4×D4 [×6], C4×Q8, C22⋊Q8 [×6], C42.C2 [×4], C4⋊Q8 [×2], C2×D8, C4×D8, D8⋊C4 [×2], C84Q8, D4⋊Q8 [×2], D4.Q8 [×4], C8.5Q8, C8⋊Q8 [×2], D43Q8 [×2], D85Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, D4○D8, D85Q8

Character table of D85Q8

 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
ρ2544-4-4000000000000000000000-222200    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000000022-2200    orthogonal lifted from D4○D8
ρ274-44-400004-40000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ284-4-440000004i-4i00000000000000000    complex lifted from D8⋊C22
ρ294-4-44000000-4i4i00000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of D85Q8
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 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 56)(41 58)(42 57)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)
(1 36 29 13)(2 37 30 14)(3 38 31 15)(4 39 32 16)(5 40 25 9)(6 33 26 10)(7 34 27 11)(8 35 28 12)(17 43 49 62)(18 44 50 63)(19 45 51 64)(20 46 52 57)(21 47 53 58)(22 48 54 59)(23 41 55 60)(24 42 56 61)
(1 57 29 46)(2 62 30 43)(3 59 31 48)(4 64 32 45)(5 61 25 42)(6 58 26 47)(7 63 27 44)(8 60 28 41)(9 24 40 56)(10 21 33 53)(11 18 34 50)(12 23 35 55)(13 20 36 52)(14 17 37 49)(15 22 38 54)(16 19 39 51)

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

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

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

Matrix representation of D85Q8 in GL8(𝔽17)

42000000
013000000
1201620000
1651610000
000016161615
00000010
00001000
00000101
,
160000000
41000000
001600000
1201610000
000016000
00000100
00001112
0000016016
,
130000000
013000000
00400000
140040000
00000400
000013000
00004448
00000131313
,
00100000
60610000
160000000
6161100000
00005204
00001511313
000022140
00001214214

G:=sub<GL(8,GF(17))| [4,0,12,16,0,0,0,0,2,13,0,5,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,16,0,0,1,0,0,0,0,16,1,0,0,0,0,0,0,15,0,0,1],[16,4,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,1,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[13,0,0,14,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,4,0,4,13,0,0,0,0,0,0,4,13,0,0,0,0,0,0,8,13],[0,6,16,6,0,0,0,0,0,0,0,16,0,0,0,0,1,6,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,15,2,12,0,0,0,0,2,1,2,14,0,0,0,0,0,13,14,2,0,0,0,0,4,13,0,14] >;

D85Q8 in GAP, Magma, Sage, TeX

D_8\rtimes_5Q_8
% in TeX

G:=Group("D8:5Q8");
// GroupNames label

G:=SmallGroup(128,2121);
// by ID

G=gap.SmallGroup(128,2121);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,346,304,4037,1027,124]);
// 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,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D85Q8 in TeX

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