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G = SD16⋊D4order 128 = 27

1st semidirect product of SD16 and D4 acting via D4/C22=C2

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

Aliases: SD165D4, C42.32C23, C4.1192+ 1+4, D424C2, C89(C2×D4), C2.49D42, Q83(C2×D4), C89D42C2, C8⋊D426C2, C83D417C2, C87D429C2, C4⋊C824C22, C4⋊C4.147D4, Q85D41C2, D4.17(C2×D4), C22⋊D826C2, D4⋊D432C2, C4⋊SD1617C2, Q8⋊D414C2, (C2×D4).306D4, (C2×D8)⋊28C22, C4⋊D48C22, (C2×C8).82C23, (C4×Q8)⋊17C22, C4.79(C22×D4), C8⋊C415C22, C2.D832C22, C22⋊Q88C22, D4.2D435C2, C22⋊SD1615C2, C4⋊C4.204C23, C22⋊C820C22, C223(C8⋊C22), (C2×C4).463C24, (C22×C8)⋊23C22, C22⋊C4.157D4, (C22×SD16)⋊6C2, C23.460(C2×D4), SD16⋊C427C2, D4⋊C435C22, C2.56(D4○SD16), Q8⋊C436C22, (C2×SD16)⋊26C22, (C4×D4).140C22, (C2×D4).203C23, C4.4D412C22, C41D4.74C22, (C2×Q8).385C23, (C22×Q8)⋊22C22, (C2×M4(2))⋊20C22, (C22×C4).317C23, C22.723(C22×D4), (C22×D4).395C22, (C2×C8⋊C22)⋊27C2, (C2×C4).587(C2×D4), C2.71(C2×C8⋊C22), (C2×C4○D4)⋊12C22, SmallGroup(128,1997)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD16⋊D4
C1C2C22C2×C4C2×D4C22×D4C22×SD16 — SD16⋊D4
C1C2C2×C4 — SD16⋊D4
C1C22C4×D4 — SD16⋊D4
C1C2C2C2×C4 — SD16⋊D4

Generators and relations for SD16⋊D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 648 in 273 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×26], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×2], D4 [×24], Q8 [×2], Q8 [×5], C23 [×2], C23 [×14], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×5], SD16 [×4], SD16 [×9], C22×C4 [×2], C22×C4 [×3], C2×D4 [×5], C2×D4 [×18], C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×3], C24 [×2], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×3], C2×SD16 [×6], C2×SD16 [×4], C8⋊C22 [×4], C22×D4 [×2], C22×D4, C22×Q8, C2×C4○D4, C89D4, SD16⋊C4, C22⋊D8, Q8⋊D4, D4⋊D4, C22⋊SD16, C4⋊SD16, D4.2D4, C87D4, C8⋊D4, C83D4, D42, Q85D4, C22×SD16, C2×C8⋊C22, SD16⋊D4
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○SD16, SD16⋊D4

Character table of SD16⋊D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11112244488822444444888444488
ρ111111111111111111111111111111    trivial
ρ211111111-11-1-111111-1-1-1-1-111111-1-1    linear of order 2
ρ31111-1-1-1-111-11111-111-1-1-11-11-1-11-11    linear of order 2
ρ41111-1-1-1-1-111-1111-11-1111-1-11-1-111-1    linear of order 2
ρ5111111-1-1-1-11-111111-1-1-1-111-1-1-1-111    linear of order 2
ρ6111111-1-11-1-11111111111-11-1-1-1-1-1-1    linear of order 2
ρ71111-1-111-1-1-1-1111-11-11111-1-111-1-11    linear of order 2
ρ81111-1-1111-111111-111-1-1-1-1-1-111-11-1    linear of order 2
ρ91111-1-1111-11-111-1-1-11-1-11-111-1-11-11    linear of order 2
ρ101111-1-111-1-1-1111-1-1-1-111-1111-1-111-1    linear of order 2
ρ11111111-1-11-1-1-111-11-1111-1-1-1111111    linear of order 2
ρ12111111-1-1-1-11111-11-1-1-1-111-11111-1-1    linear of order 2
ρ131111-1-1-1-1-111111-1-1-1-111-1-11-111-1-11    linear of order 2
ρ141111-1-1-1-111-1-111-1-1-11-1-1111-111-11-1    linear of order 2
ρ1511111111-11-1111-11-1-1-1-11-1-1-1-1-1-111    linear of order 2
ρ1611111111111-111-11-1111-11-1-1-1-1-1-1-1    linear of order 2
ρ172-22-2002-20000-22-2020000000-22000    orthogonal lifted from D4
ρ18222222002000-2-20-20-2-22000000000    orthogonal lifted from D4
ρ1922222200-2000-2-20-2022-2000000000    orthogonal lifted from D4
ρ202-22-200-220000-2220-20000000-22000    orthogonal lifted from D4
ρ212-22-2002-20000-2220-200000002-2000    orthogonal lifted from D4
ρ222222-2-200-2000-2-20202-22000000000    orthogonal lifted from D4
ρ232222-2-2002000-2-2020-22-2000000000    orthogonal lifted from D4
ρ242-22-200-220000-22-20200000002-2000    orthogonal lifted from D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000000000000-2-2002-200    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-200-2-200    complex lifted from D4○SD16

Smallest permutation representation of SD16⋊D4
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 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 26 23 12)(2 25 24 11)(3 32 17 10)(4 31 18 9)(5 30 19 16)(6 29 20 15)(7 28 21 14)(8 27 22 13)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

Matrix representation of SD16⋊D4 in GL6(𝔽17)

1600000
0160000
000010
00161612
000100
00161611
,
100000
010000
001000
0001600
00161612
0010016
,
220000
6150000
0016577
00121010
0066612
000111211
,
220000
7150000
0016577
00121010
0066612
000111211

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,16,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[2,6,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11],[2,7,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11] >;

SD16⋊D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_4
% in TeX

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

G:=SmallGroup(128,1997);
// by ID

G=gap.SmallGroup(128,1997);
# by ID

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

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

Export

Character table of SD16⋊D4 in TeX

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