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G = SD166D4order 128 = 27

2nd semidirect product of SD16 and D4 acting via D4/C22=C2

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

Aliases: SD166D4, C42.33C23, C4.1202+ 1+4, C2.50D42, (D4×Q8)⋊2C2, C89D43C2, C8.29(C2×D4), C8⋊D427C2, C4⋊C825C22, C4⋊C4.148D4, D4.18(C2×D4), C4⋊Q814C22, Q8.17(C2×D4), Q8⋊D415C2, (C2×D4).307D4, D45D4.1C2, C8.2D416C2, (C2×C8).83C23, C22⋊C4.38D4, (C4×Q8)⋊18C22, C4.80(C22×D4), C2.D833C22, C8⋊C416C22, C22⋊Q89C22, C22⋊SD1616C2, D4.D417C2, D4.7D432C2, C8.18D429C2, C4⋊C4.205C23, C22⋊C821C22, (C2×C4).464C24, Q8.D434C2, C22⋊Q1625C2, (C2×Q16)⋊27C22, (C22×SD16)⋊7C2, C23.461(C2×D4), SD16⋊C428C2, C2.57(D4○SD16), Q8⋊C437C22, (C2×D4).204C23, (C4×D4).141C22, C4⋊D4.56C22, C222(C8.C22), (C2×Q8).191C23, (C22×Q8)⋊23C22, D4⋊C4.64C22, (C2×M4(2))⋊21C22, (C22×C8).282C22, (C22×C4).318C23, (C2×SD16).47C22, C4.4D4.49C22, C22.724(C22×D4), (C22×D4).396C22, (C2×C4).588(C2×D4), (C2×C8.C22)⋊26C2, C2.70(C2×C8.C22), (C2×C4○D4).183C22, SmallGroup(128,1998)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD166D4
C1C2C22C2×C4C2×D4C22×D4C22×SD16 — SD166D4
C1C2C2×C4 — SD166D4
C1C22C4×D4 — SD166D4
C1C2C2C2×C4 — SD166D4

Generators and relations for SD166D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a5, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 520 in 247 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×16], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×17], D4 [×2], D4 [×10], Q8 [×2], Q8 [×11], C23 [×2], C23 [×7], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×4], SD16 [×9], Q16 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×3], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8 [×2], C22.D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×6], C2×SD16 [×4], C2×Q16 [×3], C8.C22 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4, C89D4, SD16⋊C4, Q8⋊D4, C22⋊SD16, C22⋊Q16, D4.7D4, D4.D4, Q8.D4, C8.18D4, C8⋊D4, C8.2D4, D45D4, D4×Q8, C22×SD16, C2×C8.C22, SD166D4
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, SD166D4

Character table of SD166D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11112244482244444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-11-11-111-11-1-11-111-11-11-11-1-11    linear of order 2
ρ31111-1-1-11-11111-1-11-11-11-11-11-11-11-1    linear of order 2
ρ4111111-1-1-1-111-1-11-1-1-1-111111111-1-1    linear of order 2
ρ5111111-11-1-1111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-1-1-1111-11-1-11-11-111-1-11-111-1    linear of order 2
ρ71111-1-1111-1111-1-11-11-1-111-1-11-11-11    linear of order 2
ρ81111111-11111-1-11-1-1-1-1-1-111-1-1-1-111    linear of order 2
ρ9111111-11-1-111-111-111-1-1-1-1-1111111    linear of order 2
ρ101111-1-1-1-1-111111-111-1-1-11-111-11-1-11    linear of order 2
ρ111111-1-1111-111-1-1-1-1-111-11-111-11-11-1    linear of order 2
ρ121111111-111111-111-1-11-1-1-1-11111-1-1    linear of order 2
ρ13111111111111-111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-1-11-11-11111-111-1-11-1-11-11-111-1    linear of order 2
ρ151111-1-1-11-1111-1-1-1-1-1111-1-11-11-11-11    linear of order 2
ρ16111111-1-1-1-1111-111-1-1111-1-1-1-1-1-111    linear of order 2
ρ172-22-20020-20-22200-200000000-20200    orthogonal lifted from D4
ρ182222220200-2-20-2-202-200000000000    orthogonal lifted from D4
ρ192-22-20020-20-22-20020000000020-200    orthogonal lifted from D4
ρ202-22-200-2020-22-200200000000-20200    orthogonal lifted from D4
ρ212222-2-20-200-2-20-2202200000000000    orthogonal lifted from D4
ρ222-22-200-2020-22200-20000000020-200    orthogonal lifted from D4
ρ232222-2-20200-2-20220-2-200000000000    orthogonal lifted from D4
ρ242222220-200-2-202-20-2200000000000    orthogonal lifted from D4
ρ254-44-40000004-400000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of SD166D4
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 10 23 32)(2 15 24 29)(3 12 17 26)(4 9 18 31)(5 14 19 28)(6 11 20 25)(7 16 21 30)(8 13 22 27)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)

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

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

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

Matrix representation of SD166D4 in GL6(𝔽17)

1600000
0160000
0012500
00121200
000055
0000125
,
100000
010000
0016000
000100
0000160
000001
,
16150000
110000
000001
000010
0001600
0016000
,
16150000
010000
000001
000010
000100
001000

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

SD166D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_6D_4
% in TeX

G:=Group("SD16:6D4");
// GroupNames label

G:=SmallGroup(128,1998);
// by ID

G=gap.SmallGroup(128,1998);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,346,2804,1411,375,172]);
// 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^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of SD166D4 in TeX

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