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G = D85D4order 128 = 27

4th semidirect product of D8 and D4 acting via D4/C4=C2

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

Aliases: D85D4, C42.440C23, C4.1272+ 1+4, C2.57D42, C8.6(C2×D4), (C4×D8)⋊32C2, C86D44C2, D45D44C2, C8⋊D431C2, C4⋊C829C22, C4⋊C4.150D4, (C4×C8)⋊30C22, D4.22(C2×D4), C22⋊D828C2, (C2×D4).164D4, C2.39(D4○D8), (C4×D4)⋊20C22, (C2×C8).89C23, (C2×Q16)⋊7C22, C4.87(C22×D4), C2.D872C22, D4.2D438C2, D4.7D438C2, C8.12D421C2, C4⋊C4.212C23, C22⋊C825C22, (C2×C4).471C24, C22⋊C4.160D4, (C2×D8).32C22, C23.313(C2×D4), C22⋊Q812C22, D4⋊C468C22, Q8⋊C441C22, (C2×SD16)⋊28C22, (C2×D4).414C23, C4⋊D4.62C22, C4.4D413C22, (C22×D4)⋊27C22, (C2×Q8).195C23, (C2×M4(2))⋊23C22, (C22×C4).321C23, C22.731(C22×D4), C2.79(D8⋊C22), (C2×C8⋊C22)⋊31C2, (C2×C4).155(C2×D4), (C2×C4○D4).187C22, SmallGroup(128,2005)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D85D4
C1C2C22C2×C4C2×D4C22×D4C2×C8⋊C22 — D85D4
C1C2C2×C4 — D85D4
C1C22C4×D4 — D85D4
C1C2C2C2×C4 — D85D4

Generators and relations for D85D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 616 in 260 conjugacy classes, 94 normal (26 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×28], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×17], Q8 [×4], C23 [×2], C23 [×14], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×4], D8 [×5], SD16 [×8], Q16, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C22⋊C4 [×2], C4×D4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16, C8⋊C22 [×8], C22×D4 [×2], C2×C4○D4 [×2], C86D4, C4×D8, C22⋊D8 [×2], D4.7D4 [×2], D4.2D4 [×2], C8⋊D4 [×2], C8.12D4, D45D4 [×2], C2×C8⋊C22 [×2], D85D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8⋊C22, D4○D8, D85D4

Character table of D85D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114444448822224448888444488
ρ111111111111111111111111111111    trivial
ρ211111-1-11-11-1-1-111-1-1-111-111-1-111-11    linear of order 2
ρ3111111111-1-11-111-1-11-1-1-11-111-1-1-11    linear of order 2
ρ411111-1-11-1-11-111111-1-1-111-1-1-1-1-111    linear of order 2
ρ51111-111-1-1111-111-1-1-11-11-1-1-1-111-11    linear of order 2
ρ61111-1-1-1-111-1-11111111-1-1-1-1111111    linear of order 2
ρ71111-111-1-1-1-1111111-1-11-1-11-1-1-1-111    linear of order 2
ρ81111-1-1-1-11-11-1-111-1-11-111-1111-1-1-11    linear of order 2
ρ91111-1-1-1-1-1-11111111-1-11-11-11111-1-1    linear of order 2
ρ101111-111-11-1-1-1-111-1-11-1111-1-1-1111-1    linear of order 2
ρ111111-1-1-1-1-11-11-111-1-1-11-111111-1-11-1    linear of order 2
ρ121111-111-1111-11111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111-1-111-111-111-1-11-1-1-1-11-1-1111-1    linear of order 2
ρ1411111111-1-1-1-111111-1-1-11-111111-1-1    linear of order 2
ρ1511111-1-1111-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ1611111111-111-1-111-1-1-111-1-1-111-1-11-1    linear of order 2
ρ172-22-2-22-22000002-2000000002-20000    orthogonal lifted from D4
ρ1822220000-22002-2-22-22-20000000000    orthogonal lifted from D4
ρ192-22-222-2-2000002-200000000-220000    orthogonal lifted from D4
ρ2022220000-2-200-2-2-2-22220000000000    orthogonal lifted from D4
ρ212-22-22-22-2000002-2000000002-20000    orthogonal lifted from D4
ρ222-22-2-2-222000002-200000000-220000    orthogonal lifted from D4
ρ23222200002-2002-2-22-2-220000000000    orthogonal lifted from D4
ρ24222200002200-2-2-2-22-2-20000000000    orthogonal lifted from D4
ρ254-44-4000000000-4400000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-400000000000000000000022-2200    orthogonal lifted from D4○D8
ρ2744-4-4000000000000000000000-222200    orthogonal lifted from D4○D8
ρ284-4-44000000004i00-4i0000000000000    complex lifted from D8⋊C22
ρ294-4-4400000000-4i004i0000000000000    complex lifted from D8⋊C22

Smallest permutation representation of D85D4
On 32 points
Generators in S32
(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)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 15 17 32)(2 16 18 25)(3 9 19 26)(4 10 20 27)(5 11 21 28)(6 12 22 29)(7 13 23 30)(8 14 24 31)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)

G:=sub<Sym(32)| (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,15,17,32)(2,16,18,25)(3,9,19,26)(4,10,20,27)(5,11,21,28)(6,12,22,29)(7,13,23,30)(8,14,24,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)>;

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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26), (1,15,17,32)(2,16,18,25)(3,9,19,26)(4,10,20,27)(5,11,21,28)(6,12,22,29)(7,13,23,30)(8,14,24,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22) );

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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)], [(1,15,17,32),(2,16,18,25),(3,9,19,26),(4,10,20,27),(5,11,21,28),(6,12,22,29),(7,13,23,30),(8,14,24,31)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22)])

Matrix representation of D85D4 in GL6(𝔽17)

1600000
0160000
000010
0000016
0001600
0016000
,
1600000
0160000
000010
000001
001000
000100
,
13150000
040000
000400
0013000
0000013
000040
,
13150000
1640000
000400
0013000
000004
0000130

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,15,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[13,16,0,0,0,0,15,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0] >;

D85D4 in GAP, Magma, Sage, TeX

D_8\rtimes_5D_4
% in TeX

G:=Group("D8:5D4");
// GroupNames label

G:=SmallGroup(128,2005);
// by ID

G=gap.SmallGroup(128,2005);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,346,2804,1411,375,172]);
// Polycyclic

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

Export

Character table of D85D4 in TeX

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