Copied to
clipboard

G = SD16⋊F5order 320 = 26·5

1st semidirect product of SD16 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD161F5, C83(C2×F5), C403(C2×C4), Q8⋊D52C4, (Q8×F5)⋊2C2, Q82(C2×F5), C40⋊C21C4, D4.D54C4, D5.D83C2, C8⋊F54C2, D4.4(C2×F5), (C2×F5).6D4, (D4×F5).2C2, C2.20(D4×F5), C5⋊(SD16⋊C4), Q8⋊F52C2, (C5×SD16)⋊1C4, D20.2(C2×C4), C10.19(C4×D4), C4⋊F5.4C22, C4.6(C22×F5), Dic102(C2×C4), D10.66(C2×D4), D20⋊C4.2C2, C20.6(C22×C4), D5⋊C8.3C22, (D5×SD16).1C2, (D4×D5).8C22, (C4×F5).3C22, (Q8×D5).5C22, D5.3(C8⋊C22), (C8×D5).25C22, (C4×D5).28C23, Dic5.4(C4○D4), D5.2(C8.C22), (C5×Q8)⋊2(C2×C4), C52C814(C2×C4), (C5×D4).4(C2×C4), SmallGroup(320,1073)

Series: Derived Chief Lower central Upper central

C1C20 — SD16⋊F5
C1C5C10D10C4×D5C4×F5D4×F5 — SD16⋊F5
C5C10C20 — SD16⋊F5
C1C2C4SD16

Generators and relations for SD16⋊F5
 G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 546 in 120 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×7], C22 [×5], C5, C8, C8 [×2], C2×C4 [×8], D4, D4 [×2], Q8, Q8 [×2], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16, SD16 [×3], C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, F5 [×4], D10, D10 [×3], C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8, C40, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C2×F5 [×2], C2×F5 [×4], C22×D5, SD16⋊C4, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D5⋊C8, C4×F5, C4×F5, C4⋊F5 [×2], C4⋊F5, C22⋊F5, D4×D5, Q8×D5, C22×F5, C8⋊F5, D5.D8, D20⋊C4, Q8⋊F5, D5×SD16, D4×F5, Q8×F5, SD16⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8⋊C22, C8.C22, C2×F5 [×3], SD16⋊C4, C22×F5, D4×F5, SD16⋊F5

Character table of SD16⋊F5

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L58A8B8C8D10A10B20A20B40A40B
 size 114552024101010101020202020204420202041681688
ρ111111111111111111111111111111    trivial
ρ211-111-11-111111-1-1-1-1-1111111-11-111    linear of order 2
ρ31111111-1-11-1-1-1-1-111-11-111-1111-1-1-1    linear of order 2
ρ411-111-111-11-1-1-111-1-111-111-11-111-1-1    linear of order 2
ρ511-111-11-1-11-1-1-11-111111-1-111-11-111    linear of order 2
ρ611111111-11-1-1-1-11-1-1-111-1-11111111    linear of order 2
ρ711-111-11111111-1111-11-1-1-1-11-111-1-1    linear of order 2
ρ81111111-1111111-1-1-111-1-1-1-1111-1-1-1    linear of order 2
ρ9111-1-1-111-i-1ii-i-i-1i-ii11-ii-1111111    linear of order 4
ρ1011-1-1-111-1-i-1ii-ii1-ii-i11-ii-11-11-111    linear of order 4
ρ11111-1-1-111i-1-i-iii-1-ii-i11i-i-1111111    linear of order 4
ρ1211-1-1-111-1i-1-i-ii-i1i-ii11i-i-11-11-111    linear of order 4
ρ13111-1-1-11-1i-1-i-iii1i-i-i1-1-ii1111-1-1-1    linear of order 4
ρ1411-1-1-1111i-1-i-ii-i-1-iii1-1-ii11-111-1-1    linear of order 4
ρ15111-1-1-11-1-i-1ii-i-i1-iii1-1i-i1111-1-1-1    linear of order 4
ρ1611-1-1-1111-i-1ii-ii-1i-i-i1-1i-i11-111-1-1    linear of order 4
ρ17220220-202-2-22-2000002000020-2000    orthogonal lifted from D4
ρ18220220-20-2-22-22000002000020-2000    orthogonal lifted from D4
ρ19220-2-20-20-2i2-2i2i2i000002000020-2000    complex lifted from C4○D4
ρ20220-2-20-202i22i-2i-2i000002000020-2000    complex lifted from C4○D4
ρ214-40-44000000000000040000-400000    orthogonal lifted from C8⋊C22
ρ2244-40004-40000000000-14000-11-11-1-1    orthogonal lifted from C2×F5
ρ23444000440000000000-14000-1-1-1-1-1-1    orthogonal lifted from F5
ρ244440004-40000000000-1-4000-1-1-1111    orthogonal lifted from C2×F5
ρ2544-4000440000000000-1-4000-11-1-111    orthogonal lifted from C2×F5
ρ264-404-4000000000000040000-400000    symplectic lifted from C8.C22, Schur index 2
ρ27880000-800000000000-20000-202000    orthogonal lifted from D4×F5
ρ288-80000000000000000-200002000--10-10    complex faithful
ρ298-80000000000000000-200002000-10--10    complex faithful

Smallest permutation representation of SD16⋊F5
On 40 points
Generators in S40
(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)(33 34 35 36 37 38 39 40)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)
(1 26 19 12 35)(2 27 20 13 36)(3 28 21 14 37)(4 29 22 15 38)(5 30 23 16 39)(6 31 24 9 40)(7 32 17 10 33)(8 25 18 11 34)
(2 6)(4 8)(9 27 24 36)(10 32 17 33)(11 29 18 38)(12 26 19 35)(13 31 20 40)(14 28 21 37)(15 25 22 34)(16 30 23 39)

G:=sub<Sym(40)| (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)(33,34,35,36,37,38,39,40), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38), (1,26,19,12,35)(2,27,20,13,36)(3,28,21,14,37)(4,29,22,15,38)(5,30,23,16,39)(6,31,24,9,40)(7,32,17,10,33)(8,25,18,11,34), (2,6)(4,8)(9,27,24,36)(10,32,17,33)(11,29,18,38)(12,26,19,35)(13,31,20,40)(14,28,21,37)(15,25,22,34)(16,30,23,39)>;

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)(33,34,35,36,37,38,39,40), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38), (1,26,19,12,35)(2,27,20,13,36)(3,28,21,14,37)(4,29,22,15,38)(5,30,23,16,39)(6,31,24,9,40)(7,32,17,10,33)(8,25,18,11,34), (2,6)(4,8)(9,27,24,36)(10,32,17,33)(11,29,18,38)(12,26,19,35)(13,31,20,40)(14,28,21,37)(15,25,22,34)(16,30,23,39) );

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),(33,34,35,36,37,38,39,40)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,21),(18,24),(20,22),(25,31),(27,29),(28,32),(33,37),(34,40),(36,38)], [(1,26,19,12,35),(2,27,20,13,36),(3,28,21,14,37),(4,29,22,15,38),(5,30,23,16,39),(6,31,24,9,40),(7,32,17,10,33),(8,25,18,11,34)], [(2,6),(4,8),(9,27,24,36),(10,32,17,33),(11,29,18,38),(12,26,19,35),(13,31,20,40),(14,28,21,37),(15,25,22,34),(16,30,23,39)])

Matrix representation of SD16⋊F5 in GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
00000010
0000000-1
00000-100
0000-1000
,
10000000
01000000
00100000
00010000
00001000
00000-100
00000001
00000010
,
000-10000
100-10000
010-10000
001-10000
00001000
00000100
00000010
00000001
,
00100000
10000000
00010000
01000000
00001000
00000100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;

SD16⋊F5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes F_5
% in TeX

G:=Group("SD16:F5");
// GroupNames label

G:=SmallGroup(320,1073);
// by ID

G=gap.SmallGroup(320,1073);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of SD16⋊F5 in TeX

׿
×
𝔽