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G = SD167D4order 128 = 27

3rd semidirect product of SD16 and D4 acting via D4/C22=C2

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

Aliases: SD167D4, C42.35C23, C4.1222+ 1+4, C2.52D42, C89D45C2, C8.31(C2×D4), D45D43C2, C4⋊D834C2, C83D418C2, C87D430C2, C8⋊D428C2, C4⋊C827C22, C4⋊C4.356D4, Q86D42C2, D4.20(C2×D4), Q8.18(C2×D4), D4⋊D434C2, C22⋊D827C2, (C2×D4).161D4, C2.38(D4○D8), C4⋊D49C22, (C2×C8).85C23, C22⋊C4.40D4, (C4×Q8)⋊19C22, C4.82(C22×D4), C8⋊C418C22, C2.D834C22, D4.7D434C2, C4⋊C4.207C23, C41D410C22, C22⋊C823C22, (C2×C4).466C24, Q8.D435C2, (C22×C8)⋊25C22, (C2×Q16)⋊46C22, (C2×D8).79C22, C23.100(C2×D4), SD16⋊C429C2, D4⋊C437C22, Q8⋊C439C22, (C2×SD16)⋊27C22, (C4×D4).142C22, (C2×D4).206C23, (C2×Q8).386C23, C22⋊Q8.56C22, C4.4D4.51C22, C22.726(C22×D4), C2.76(D8⋊C22), (C22×C4).1118C23, (C22×D4).398C22, (C2×M4(2)).101C22, (C2×C4○D8)⋊24C2, (C2×C8⋊C22)⋊29C2, (C2×C4).590(C2×D4), (C2×C4○D4)⋊14C22, SmallGroup(128,2000)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD167D4
C1C2C22C2×C4C2×D4C2×C4○D4C2×C4○D8 — SD167D4
C1C2C2×C4 — SD167D4
C1C22C4×D4 — SD167D4
C1C2C2C2×C4 — SD167D4

Generators and relations for SD167D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 584 in 255 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×23], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×2], D4 [×22], Q8 [×2], Q8 [×3], C23 [×2], C23 [×9], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×7], SD16 [×4], SD16 [×5], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×5], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×9], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4 [×3], C4⋊D4 [×3], C22⋊Q8, C22.D4, C4.4D4, C41D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4○D8 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×3], C89D4, SD16⋊C4, C22⋊D8, D4⋊D4 [×2], D4.7D4, C4⋊D8, Q8.D4, C87D4, C8⋊D4, C83D4, D45D4, Q86D4, C2×C4○D8, C2×C8⋊C22, SD167D4
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, SD167D4

Character table of SD167D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114444888222244444888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-1-1-1-11111-1111-1-1-1-1111111    linear of order 2
ρ3111111-1-11-11111111111-11-1-1-1-1-1-1-1    linear of order 2
ρ411111111-11-11111-1111-11-11-1-1-1-1-1-1    linear of order 2
ρ511111-1-1-1-11111111-1-1-11-1-11-1-1-1-111    linear of order 2
ρ611111-1111-1-11111-1-1-1-1-111-1-1-1-1-111    linear of order 2
ρ711111-111-1-1111111-1-1-111-1-11111-1-1    linear of order 2
ρ811111-1-1-111-11111-1-1-1-1-1-1111111-1-1    linear of order 2
ρ91111-11-1-1-1-11-111-1-11-1-1-11111-1-111-1    linear of order 2
ρ101111-111111-1-111-111-1-11-1-1-11-1-111-1    linear of order 2
ρ111111-1111-111-111-1-11-1-1-1-11-1-111-1-11    linear of order 2
ρ121111-11-1-11-1-1-111-111-1-111-11-111-1-11    linear of order 2
ρ131111-1-1111-11-111-1-1-111-1-1-11-111-11-1    linear of order 2
ρ141111-1-1-1-1-11-1-111-11-111111-1-111-11-1    linear of order 2
ρ151111-1-1-1-1111-111-1-1-111-11-1-11-1-11-11    linear of order 2
ρ161111-1-111-1-1-1-111-11-1111-1111-1-11-11    linear of order 2
ρ172-22-2002-200002-20-20002000-200200    orthogonal lifted from D4
ρ182-22-200-2200002-202000-2000-200200    orthogonal lifted from D4
ρ192222-22000002-2-220-2-220000000000    orthogonal lifted from D4
ρ202-22-200-2200002-20-20002000200-200    orthogonal lifted from D4
ρ212-22-2002-200002-202000-2000200-200    orthogonal lifted from D4
ρ2222222-200000-2-2-2-202-220000000000    orthogonal lifted from D4
ρ2322222200000-2-2-2-20-22-20000000000    orthogonal lifted from D4
ρ242222-2-2000002-2-22022-20000000000    orthogonal lifted from D4
ρ254-44-400000000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-400000000000000000000-2222000    orthogonal lifted from D4○D8
ρ2744-4-40000000000000000000022-22000    orthogonal lifted from D4○D8
ρ284-4-440000000-4i004i00000000000000    complex lifted from D8⋊C22
ρ294-4-4400000004i00-4i00000000000000    complex lifted from D8⋊C22

Smallest permutation representation of SD167D4
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 10)(2 13)(3 16)(4 11)(5 14)(6 9)(7 12)(8 15)(17 30)(18 25)(19 28)(20 31)(21 26)(22 29)(23 32)(24 27)
(1 24 14 27)(2 23 15 26)(3 22 16 25)(4 21 9 32)(5 20 10 31)(6 19 11 30)(7 18 12 29)(8 17 13 28)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)

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

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

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

Matrix representation of SD167D4 in GL6(𝔽17)

1600000
0160000
0000314
000033
0014300
00141400
,
100000
010000
0016000
000100
0000160
000001
,
16150000
110000
000001
000010
000100
001000
,
1600000
110000
001000
0001600
0000160
000001

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

SD167D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_7D_4
% in TeX

G:=Group("SD16:7D4");
// GroupNames label

G:=SmallGroup(128,2000);
// by ID

G=gap.SmallGroup(128,2000);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,346,2804,1411,375,172]);
// Polycyclic

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

Export

Character table of SD167D4 in TeX

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