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G = D87D4order 128 = 27

1st semidirect product of D8 and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D87D4, C222D16, C23.44D8, (C2×D16)⋊1C2, C87D41C2, C8.60(C2×D4), (C2×C8).61D4, (C2×C4).30D8, C2.4(C2×D16), C2.D164C2, C22⋊C165C2, (C2×C16)⋊1C22, (C22×D8)⋊4C2, (C2×D8)⋊1C22, C2.D81C22, C4.16C22≀C2, C4.8(C8⋊C22), C22.94(C2×D8), C2.6(C16⋊C22), (C2×C8).508C23, (C22×C4).343D4, C2.24(C22⋊D8), (C22×C8).125C22, (C2×C4).776(C2×D4), SmallGroup(128,916)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D87D4
C1C2C4C2×C4C2×C8C22×C8C22×D8 — D87D4
C1C2C4C2×C8 — D87D4
C1C22C22×C4C22×C8 — D87D4
C1C2C2C2C2C4C4C2×C8 — D87D4

Generators and relations for D87D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=a5b, dcd=c-1 >

Subgroups: 452 in 133 conjugacy classes, 36 normal (20 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], C8 [×2], C8, C2×C4 [×2], C2×C4 [×3], D4 [×14], C23, C23 [×11], C16 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×2], D8 [×4], D8 [×8], C22×C4, C2×D4 [×11], C24, D4⋊C4, C2.D8, C2×C16 [×2], D16 [×4], C4⋊D4, C22×C8, C2×D8, C2×D8 [×2], C2×D8 [×5], C22×D4, C22⋊C16, C2.D16 [×2], C87D4, C2×D16 [×2], C22×D8, D87D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], D16 [×2], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C2×D16, C16⋊C22, D87D4

Character table of D87D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111228888162241622224444444444
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-11111-1-11111-1-1-1-11-1111-1    linear of order 2
ρ31111111111-1111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-11-11-111-111111-1-111-11-1-1-11    linear of order 2
ρ51111-1-11-11-1-111-111111-1-1-1-11-1111-1    linear of order 2
ρ6111111-1-1-1-1-1111-111111111111111    linear of order 2
ρ71111-1-11-11-1111-1-11111-1-111-11-1-1-11    linear of order 2
ρ8111111-1-1-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92-2-2200-202002-2002-22-20000000000    orthogonal lifted from D4
ρ102-2-2200020-202-200-22-220000000000    orthogonal lifted from D4
ρ112-2-220020-2002-2002-22-20000000000    orthogonal lifted from D4
ρ12222222000002220-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ132222-2-20000022-20-2-2-2-22200000000    orthogonal lifted from D4
ρ142-2-22000-20202-200-22-220000000000    orthogonal lifted from D4
ρ1522222200000-2-2-20000000-2-2-222-222    orthogonal lifted from D8
ρ1622222200000-2-2-20000000222-2-22-2-2    orthogonal lifted from D8
ρ172222-2-200000-2-22000000022-2-22-22-2    orthogonal lifted from D8
ρ182222-2-200000-2-220000000-2-222-22-22    orthogonal lifted from D8
ρ192-22-22-2000000000-222-2-22ζ167161671616716165163ζ165163ζ16716165163ζ165163    orthogonal lifted from D16
ρ202-22-22-2000000000-222-2-2216716ζ16716ζ16716ζ16516316516316716ζ165163165163    orthogonal lifted from D16
ρ212-22-22-20000000002-2-222-2165163ζ165163ζ16516316716ζ1671616516316716ζ16716    orthogonal lifted from D16
ρ222-22-22-20000000002-2-222-2ζ165163165163165163ζ1671616716ζ165163ζ1671616716    orthogonal lifted from D16
ρ232-22-2-22000000000-222-22-2ζ1671616716ζ1671616516316516316716ζ165163ζ165163    orthogonal lifted from D16
ρ242-22-2-220000000002-2-22-22ζ165163165163ζ165163ζ16716ζ167161651631671616716    orthogonal lifted from D16
ρ252-22-2-220000000002-2-22-22165163ζ1651631651631671616716ζ165163ζ16716ζ16716    orthogonal lifted from D16
ρ262-22-2-22000000000-222-22-216716ζ1671616716ζ165163ζ165163ζ16716165163165163    orthogonal lifted from D16
ρ274-4-440000000-440000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ2944-4-4000000000002222-22-220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of D87D4
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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)
(1 30 17 12)(2 29 18 11)(3 28 19 10)(4 27 20 9)(5 26 21 16)(6 25 22 15)(7 32 23 14)(8 31 24 13)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25), (1,30,17,12)(2,29,18,11)(3,28,19,10)(4,27,20,9)(5,26,21,16)(6,25,22,15)(7,32,23,14)(8,31,24,13), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25), (1,30,17,12)(2,29,18,11)(3,28,19,10)(4,27,20,9)(5,26,21,16)(6,25,22,15)(7,32,23,14)(8,31,24,13), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25)], [(1,30,17,12),(2,29,18,11),(3,28,19,10),(4,27,20,9),(5,26,21,16),(6,25,22,15),(7,32,23,14),(8,31,24,13)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31)])

Matrix representation of D87D4 in GL4(𝔽17) generated by

31400
3300
00160
00016
,
01600
16000
00160
0001
,
6400
41100
0001
00160
,
6400
41100
0001
0010
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,16,0,0,0,0,1],[6,4,0,0,4,11,0,0,0,0,0,16,0,0,1,0],[6,4,0,0,4,11,0,0,0,0,0,1,0,0,1,0] >;

D87D4 in GAP, Magma, Sage, TeX

D_8\rtimes_7D_4
% in TeX

G:=Group("D8:7D4");
// GroupNames label

G:=SmallGroup(128,916);
// by ID

G=gap.SmallGroup(128,916);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,1123,570,360,4037,2028,124]);
// Polycyclic

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

Export

Character table of D87D4 in TeX

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