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G = D810D4order 128 = 27

4th semidirect product of D8 and D4 acting via D4/C22=C2

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

Aliases: D810D4, C42.34C23, C4.1212+ 1+4, C2.51D42, C89D44C2, C8.30(C2×D4), D45D42C2, D46D42C2, C88D418C2, C4⋊C826C22, C4⋊C4.355D4, D4.19(C2×D4), C4⋊Q815C22, D8⋊C420C2, D4⋊D433C2, (C2×D4).160D4, C8.D419C2, C8.2D417C2, (C4×D4)⋊18C22, (C2×C8).84C23, C22⋊C4.39D4, C4.81(C22×D4), C23.99(C2×D4), C4.Q823C22, C8⋊C417C22, D4.7D433C2, D4.D418C2, C22⋊SD1617C2, D4.2D436C2, C4⋊C4.206C23, C22⋊C822C22, (C2×C4).465C24, (C22×C8)⋊24C22, (C2×Q16)⋊28C22, C22⋊Q810C22, D4⋊C436C22, C2.58(D4○SD16), Q8⋊C438C22, (C2×SD16)⋊80C22, (C2×D4).205C23, (C2×D8).165C22, C4⋊D4.57C22, (C2×Q8).192C23, C4.4D4.50C22, C22.725(C22×D4), C2.75(D8⋊C22), (C22×C4).1117C23, (C22×D4).397C22, (C2×M4(2)).100C22, (C2×C4○D8)⋊23C2, (C2×C8⋊C22)⋊28C2, (C2×C4).589(C2×D4), (C2×C4○D4)⋊13C22, SmallGroup(128,1999)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D810D4
C1C2C22C2×C4C2×D4C2×C4○D4C2×C4○D8 — D810D4
C1C2C2×C4 — D810D4
C1C22C4×D4 — D810D4
C1C2C2C2×C4 — D810D4

Generators and relations for D810D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >

Subgroups: 544 in 250 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×21], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×18], D4 [×4], D4 [×14], Q8 [×6], C23 [×2], C23 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], D8 [×2], SD16 [×9], Q16 [×3], C22×C4 [×2], C22×C4 [×5], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×3], C4○D4 [×9], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×3], C22≀C2, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C22.D4 [×3], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×5], C2×Q16 [×2], C4○D8 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×3], C89D4, D8⋊C4, D4⋊D4, C22⋊SD16, D4.7D4 [×2], D4.D4, D4.2D4, C88D4, C8.D4, C8.2D4, D45D4, D46D4, C2×C4○D8, C2×C8⋊C22, D810D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8⋊C22, D4○SD16, D810D4

Character table of D810D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114444448222244488888444488
ρ111111111111111111111111111111    trivial
ρ21111-11111-1111111111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-11-1-111-11-1-11-1-111-11-11-11    linear of order 2
ρ41111-11-1-1-1-1-1-111-11-1-1111-111-11-11-1    linear of order 2
ρ511111-111-11-1-111-1-11111-1-1-11-11-1-11    linear of order 2
ρ61111-1-111-1-1-1-111-1-1111-111-1-11-111-1    linear of order 2
ρ711111-1-1-11111111-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ81111-1-1-1-11-111111-1-1-111-11-11111-1-1    linear of order 2
ρ911111111-111-111-11-1-1-1-1-11-11-11-11-1    linear of order 2
ρ101111-1111-1-11-111-11-1-1-111-1-1-11-11-11    linear of order 2
ρ11111111-1-111-11111111-1111-1-1-1-1-1-1-1    linear of order 2
ρ121111-11-1-11-1-11111111-1-1-1-1-1111111    linear of order 2
ρ1311111-11111-11111-1-1-1-1-11-111111-1-1    linear of order 2
ρ141111-1-1111-1-11111-1-1-1-11-111-1-1-1-111    linear of order 2
ρ1511111-1-1-1-111-111-1-111-11-1-11-11-111-1    linear of order 2
ρ161111-1-1-1-1-1-11-111-1-111-1-11111-11-1-11    linear of order 2
ρ172-22-2-20-220200-22000000000-202000    orthogonal lifted from D4
ρ182-22-2202-20-200-22000000000-202000    orthogonal lifted from D4
ρ1922220-200-2002-2-2222-200000000000    orthogonal lifted from D4
ρ202-22-220-220-200-2200000000020-2000    orthogonal lifted from D4
ρ2122220-200200-2-2-2-22-2200000000000    orthogonal lifted from D4
ρ222-22-2-202-20200-2200000000020-2000    orthogonal lifted from D4
ρ2322220200-2002-2-22-2-2200000000000    orthogonal lifted from D4
ρ2422220200200-2-2-2-2-22-200000000000    orthogonal lifted from D4
ρ254-44-4000000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440000000-4i004i00000000000000    complex lifted from D8⋊C22
ρ274-4-4400000004i00-4i00000000000000    complex lifted from D8⋊C22
ρ2844-4-400000000000000000000-2-202-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002-20-2-200    complex lifted from D4○SD16

Smallest permutation representation of D810D4
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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)
(1 17 29 13)(2 20 30 16)(3 23 31 11)(4 18 32 14)(5 21 25 9)(6 24 26 12)(7 19 27 15)(8 22 28 10)
(2 4)(3 7)(6 8)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)(26 28)(27 31)(30 32)

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

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

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

Matrix representation of D810D4 in GL6(𝔽17)

100000
010000
000007
000057
0001000
00121000
,
100000
010000
0000010
000050
000700
0012000
,
0160000
100000
0000115
0000016
0011500
0001600
,
1600000
010000
001000
0011600
0000160
0000161

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,5,0,0,0,0,7,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,5,0,0,0,0,10,0,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,1,0,0,0,0,0,15,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;

D810D4 in GAP, Magma, Sage, TeX

D_8\rtimes_{10}D_4
% in TeX

G:=Group("D8:10D4");
// GroupNames label

G:=SmallGroup(128,1999);
// by ID

G=gap.SmallGroup(128,1999);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,346,2804,1411,375,172]);
// 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=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D810D4 in TeX

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