Copied to
clipboard

G = D10⋊C8order 160 = 25·5

2nd semidirect product of D10 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102C8, Dic5.20D4, C10.3M4(2), (C2×C4).3F5, C51(C22⋊C8), C10.3(C2×C8), (C2×C20).3C4, C2.4(D5⋊C8), C2.3(C4.F5), (C22×D5).6C4, C2.1(C22⋊F5), C22.11(C2×F5), C10.2(C22⋊C4), (C2×Dic5).51C22, (C2×C5⋊C8)⋊1C2, (C2×C4×D5).8C2, (C2×C10).6(C2×C4), SmallGroup(160,78)

Series: Derived Chief Lower central Upper central

C1C10 — D10⋊C8
C1C5C10Dic5C2×Dic5C2×C5⋊C8 — D10⋊C8
C5C10 — D10⋊C8
C1C22C2×C4

Generators and relations for D10⋊C8
 G = < a,b,c | a10=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a7b >

10C2
10C2
2C4
5C4
5C22
5C22
5C4
10C22
10C22
2D5
2D5
5C23
5C2×C4
10C8
10C2×C4
10C8
10C2×C4
2D10
2C20
2D10
5C2×C8
5C22×C4
5C2×C8
2C5⋊C8
2C4×D5
2C5⋊C8
2C4×D5
5C22⋊C8

Character table of D10⋊C8

 class 12A2B2C2D2E4A4B4C4D4E4F58A8B8C8D8E8F8G8H10A10B10C20A20B20C20D
 size 11111010225555410101010101010104444444
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-111111-1111-1-1-11111-1-1-1-1    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ41111-1-1-1-1111111-1-1-1111-1111-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-1-11-i-iiiii-i-i111-1-1-1-1    linear of order 4
ρ6111111-1-1-1-1-1-11ii-i-i-i-iii111-1-1-1-1    linear of order 4
ρ71111-1-111-1-1-1-11-ii-i-iii-ii1111111    linear of order 4
ρ81111-1-111-1-1-1-11i-iii-i-ii-i1111111    linear of order 4
ρ91-11-1-11-iii-ii-i1ζ87ζ87ζ8ζ85ζ85ζ8ζ83ζ83-1-11i-i-ii    linear of order 8
ρ101-11-1-11-iii-ii-i1ζ83ζ83ζ85ζ8ζ8ζ85ζ87ζ87-1-11i-i-ii    linear of order 8
ρ111-11-1-11i-i-ii-ii1ζ8ζ8ζ87ζ83ζ83ζ87ζ85ζ85-1-11-iii-i    linear of order 8
ρ121-11-1-11i-i-ii-ii1ζ85ζ85ζ83ζ87ζ87ζ83ζ8ζ8-1-11-iii-i    linear of order 8
ρ131-11-11-1i-ii-ii-i1ζ83ζ87ζ8ζ85ζ8ζ85ζ87ζ83-1-11-iii-i    linear of order 8
ρ141-11-11-1-ii-ii-ii1ζ8ζ85ζ83ζ87ζ83ζ87ζ85ζ8-1-11i-i-ii    linear of order 8
ρ151-11-11-1-ii-ii-ii1ζ85ζ8ζ87ζ83ζ87ζ83ζ8ζ85-1-11i-i-ii    linear of order 8
ρ161-11-11-1i-ii-ii-i1ζ87ζ83ζ85ζ8ζ85ζ8ζ83ζ87-1-11-iii-i    linear of order 8
ρ1722-2-20000-222-22000000002-2-20000    orthogonal lifted from D4
ρ1822-2-200002-2-222000000002-2-20000    orthogonal lifted from D4
ρ192-2-2200002i2i-2i-2i200000000-22-20000    complex lifted from M4(2)
ρ202-2-220000-2i-2i2i2i200000000-22-20000    complex lifted from M4(2)
ρ21444400440000-100000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ22444400-4-40000-100000000-1-1-11111    orthogonal lifted from C2×F5
ρ2344-4-400000000-100000000-111-55-55    orthogonal lifted from C22⋊F5
ρ2444-4-400000000-100000000-1115-55-5    orthogonal lifted from C22⋊F5
ρ254-44-4004i-4i0000-10000000011-1i-i-ii    complex lifted from D5⋊C8, Schur index 2
ρ264-44-400-4i4i0000-10000000011-1-iii-i    complex lifted from D5⋊C8, Schur index 2
ρ274-4-4400000000-1000000001-11-5-5--5--5    complex lifted from C4.F5
ρ284-4-4400000000-1000000001-11--5--5-5-5    complex lifted from C4.F5

Smallest permutation representation of D10⋊C8
On 80 points
Generators in S80
(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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 73)(12 72)(13 71)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 50)(38 49)(39 48)(40 47)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
(1 14 46 65 21 76 32 58)(2 11 45 68 22 73 31 51)(3 18 44 61 23 80 40 54)(4 15 43 64 24 77 39 57)(5 12 42 67 25 74 38 60)(6 19 41 70 26 71 37 53)(7 16 50 63 27 78 36 56)(8 13 49 66 28 75 35 59)(9 20 48 69 29 72 34 52)(10 17 47 62 30 79 33 55)

G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,73)(12,72)(13,71)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,14,46,65,21,76,32,58)(2,11,45,68,22,73,31,51)(3,18,44,61,23,80,40,54)(4,15,43,64,24,77,39,57)(5,12,42,67,25,74,38,60)(6,19,41,70,26,71,37,53)(7,16,50,63,27,78,36,56)(8,13,49,66,28,75,35,59)(9,20,48,69,29,72,34,52)(10,17,47,62,30,79,33,55)>;

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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,73)(12,72)(13,71)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,14,46,65,21,76,32,58)(2,11,45,68,22,73,31,51)(3,18,44,61,23,80,40,54)(4,15,43,64,24,77,39,57)(5,12,42,67,25,74,38,60)(6,19,41,70,26,71,37,53)(7,16,50,63,27,78,36,56)(8,13,49,66,28,75,35,59)(9,20,48,69,29,72,34,52)(10,17,47,62,30,79,33,55) );

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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,73),(12,72),(13,71),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,50),(38,49),(39,48),(40,47),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67)], [(1,14,46,65,21,76,32,58),(2,11,45,68,22,73,31,51),(3,18,44,61,23,80,40,54),(4,15,43,64,24,77,39,57),(5,12,42,67,25,74,38,60),(6,19,41,70,26,71,37,53),(7,16,50,63,27,78,36,56),(8,13,49,66,28,75,35,59),(9,20,48,69,29,72,34,52),(10,17,47,62,30,79,33,55)])

D10⋊C8 is a maximal subgroup of
D10.1D8  D10.1Q16  D10.SD16  D10.Q16  C42.6F5  C42.12F5  C42.15F5  C42.7F5  C5⋊C88D4  C5⋊C8⋊D4  D10⋊M4(2)  Dic5⋊M4(2)  D202C8  D102M4(2)  C20⋊M4(2)  C4⋊C4.7F5  D10.11M4(2)  D109M4(2)  D1010M4(2)  (C2×D4).7F5  (C2×Q8).5F5  D30⋊C8  C30.7M4(2)
D10⋊C8 is a maximal quotient of
D20⋊C8  Dic101C8  D10⋊C16  C20.10M4(2)  D20.C8  (C22×C4).F5  C5⋊(C23⋊C8)  C10.(C4⋊C8)  D30⋊C8  C30.7M4(2)

Matrix representation of D10⋊C8 in GL8(𝔽41)

400000000
040000000
004000000
000400000
000000401
000000400
000010400
000001400
,
400000000
01000000
00100000
0040400000
000001400
000010400
000000400
000000401
,
01000000
10000000
0040390000
00510000
00001427230
00003727014
00001402737
00000232714

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,14,37,14,0,0,0,0,0,27,27,0,23,0,0,0,0,23,0,27,27,0,0,0,0,0,14,37,14] >;

D10⋊C8 in GAP, Magma, Sage, TeX

D_{10}\rtimes C_8
% in TeX

G:=Group("D10:C8");
// GroupNames label

G:=SmallGroup(160,78);
// by ID

G=gap.SmallGroup(160,78);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,86,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^10=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D10⋊C8 in TeX
Character table of D10⋊C8 in TeX

׿
×
𝔽