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G = D89D4order 128 = 27

3rd semidirect product of D8 and D4 acting via D4/C22=C2

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

Aliases: D89D4, C42.31C23, C4.1182+ 1+4, D423C2, C88(C2×D4), C2.48D42, D43(C2×D4), C89D41C2, D45D41C2, C4⋊D833C2, C83D416C2, C82D419C2, C88D417C2, C4⋊C823C22, C4⋊C4.146D4, D8⋊C419C2, D4⋊D431C2, C22⋊D825C2, (C2×D4).305D4, C2.37(D4○D8), (C2×D8)⋊27C22, (C4×D4)⋊17C22, (C22×D8)⋊20C2, C4⋊D47C22, C41D49C22, (C2×C8).81C23, C4.78(C22×D4), C8⋊C414C22, C4.Q822C22, C22⋊Q87C22, C22⋊SD1614C2, D4.2D434C2, C4⋊C4.203C23, C22⋊C819C22, C222(C8⋊C22), (C2×C4).462C24, C22⋊C4.156D4, (C22×C8)⋊22C22, Q8⋊C47C22, C23.459(C2×D4), D4⋊C434C22, (C2×SD16)⋊25C22, (C2×D4).202C23, C4.4D411C22, (C22×D4)⋊26C22, (C2×Q8).190C23, (C2×M4(2))⋊19C22, (C22×C4).316C23, C22.722(C22×D4), (C2×C8⋊C22)⋊26C2, (C2×C4).586(C2×D4), C2.70(C2×C8⋊C22), (C2×C4○D4)⋊11C22, SmallGroup(128,1996)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D89D4
C1C2C22C2×C4C2×D4C22×D4C22×D8 — D89D4
C1C2C2×C4 — D89D4
C1C22C4×D4 — D89D4
C1C2C2C2×C4 — D89D4

Generators and relations for D89D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=dad=a3, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 720 in 284 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×10], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×34], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×12], D4 [×4], D4 [×26], Q8 [×2], C23 [×2], C23 [×20], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4, C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], D8 [×9], SD16 [×5], C22×C4 [×2], C22×C4 [×3], C2×D4 [×6], C2×D4 [×22], C2×Q8, C4○D4 [×3], C24 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×5], Q8⋊C4, C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4 [×3], C22≀C2 [×3], C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×6], C2×D8 [×4], C2×SD16 [×3], C8⋊C22 [×4], C22×D4 [×3], C22×D4, C2×C4○D4, C89D4, D8⋊C4, C22⋊D8 [×2], D4⋊D4, C22⋊SD16, C4⋊D8, D4.2D4, C88D4, C82D4, C83D4, D42, D45D4, C22×D8, C2×C8⋊C22, D89D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8⋊C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8⋊C22, D4○D8, D89D4

Character table of D89D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F
 size 11112244444888224444888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-1-11-11-111-1-111-1111-11-11-1    linear of order 2
ρ31111111-111-1-11-111111111-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-1-1-1-1-111111-1-111-11-1-11-11-11    linear of order 2
ρ5111111-11-1111-11111111-1-11-1-1-1-1-1-1    linear of order 2
ρ61111-1-1111-11-1-1-111-1-1111-11-11-11-11    linear of order 2
ρ7111111-1-1-11-1-1-1-1111111-1-1-1111111    linear of order 2
ρ81111-1-11-11-1-11-1111-1-1111-1-11-11-11-1    linear of order 2
ρ9111111111-1111-1111-1-1-1-1-1-11111-1-1    linear of order 2
ρ101111-1-1-11-111-11111-11-1-11-1-11-11-1-11    linear of order 2
ρ111111111-11-1-1-111111-1-1-1-1-11-1-1-1-111    linear of order 2
ρ121111-1-1-1-1-11-111-111-11-1-11-11-11-111-1    linear of order 2
ρ13111111-11-1-111-1-1111-1-1-111-1-1-1-1-111    linear of order 2
ρ141111-1-111111-1-1111-11-1-1-11-1-11-111-1    linear of order 2
ρ15111111-1-1-1-1-1-1-11111-1-1-11111111-1-1    linear of order 2
ρ161111-1-11-111-11-1-111-11-1-1-1111-11-1-11    linear of order 2
ρ172-22-200-2-2202000-220000000020-200    orthogonal lifted from D4
ρ182-22-2002-2-202000-2200000000-20200    orthogonal lifted from D4
ρ192222-2-2000-20000-2-222-22000000000    orthogonal lifted from D4
ρ202222-2-200020000-2-22-22-2000000000    orthogonal lifted from D4
ρ212-22-200-2220-2000-2200000000-20200    orthogonal lifted from D4
ρ2222222200020000-2-2-2-2-22000000000    orthogonal lifted from D4
ρ23222222000-20000-2-2-222-2000000000    orthogonal lifted from D4
ρ242-22-20022-20-2000-220000000020-200    orthogonal lifted from D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000000004-40000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000000000000220-22000    orthogonal lifted from D4○D8
ρ2944-4-40000000000000000000-22022000    orthogonal lifted from D4○D8

Smallest permutation representation of D89D4
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 20)(2 19)(3 18)(4 17)(5 24)(6 23)(7 22)(8 21)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 30 21 13)(2 25 22 16)(3 28 23 11)(4 31 24 14)(5 26 17 9)(6 29 18 12)(7 32 19 15)(8 27 20 10)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)

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

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

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

Matrix representation of D89D4 in GL6(𝔽17)

100000
010000
000600
0014600
0000116
0000140
,
100000
010000
0001100
0014000
0000611
0000311
,
0160000
100000
000010
000001
0016000
0001600
,
0160000
1600000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,6,6,0,0,0,0,0,0,11,14,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,0,0,0,0,6,3,0,0,0,0,11,11],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

D89D4 in GAP, Magma, Sage, TeX

D_8\rtimes_9D_4
% in TeX

G:=Group("D8:9D4");
// GroupNames label

G:=SmallGroup(128,1996);
// by ID

G=gap.SmallGroup(128,1996);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D89D4 in TeX

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