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G = C8⋊D5order 80 = 24·5

3rd semidirect product of C8 and D5 acting via D5/C5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D5, C404C2, C53M4(2), D10.1C4, C4.13D10, Dic5.1C4, C20.13C22, C52C84C2, C2.3(C4×D5), C10.9(C2×C4), (C4×D5).2C2, SmallGroup(80,5)

Series: Derived Chief Lower central Upper central

C1C10 — C8⋊D5
C1C5C10C20C4×D5 — C8⋊D5
C5C10 — C8⋊D5
C1C4C8

Generators and relations for C8⋊D5
 G = < a,b,c | a8=b5=c2=1, ab=ba, cac=a5, cbc=b-1 >

10C2
5C22
5C4
2D5
5C2×C4
5C8
5M4(2)

Character table of C8⋊D5

 class 12A2B4A4B4C5A5B8A8B8C8D10A10B20A20B20C20D40A40B40C40D40E40F40G40H
 size 111011102222101022222222222222
ρ111111111111111111111111111    trivial
ρ211-111-11111-1-111111111111111    linear of order 2
ρ311111111-1-1-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111-111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111-1-1-111i-i-ii11-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ611-1-1-1111i-ii-i11-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ7111-1-1-111-iii-i11-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ811-1-1-1111-ii-ii11-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ9220220-1-5/2-1+5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ10220220-1+5/2-1-5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ11220220-1-5/2-1+5/22200-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ12220220-1+5/2-1-5/22200-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ132-20-2i2i0220000-2-2-2i2i-2i2i00000000    complex lifted from M4(2)
ρ142-202i-2i0220000-2-22i-2i2i-2i00000000    complex lifted from M4(2)
ρ15220-2-20-1+5/2-1-5/2-2i2i00-1-5/2-1+5/21-5/21+5/21+5/21-5/2ζ4ζ534ζ52ζ4ζ544ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52    complex lifted from C4×D5
ρ16220-2-20-1-5/2-1+5/22i-2i00-1+5/2-1-5/21+5/21-5/21-5/21+5/2ζ43ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ4ζ544ζ5    complex lifted from C4×D5
ρ17220-2-20-1+5/2-1-5/22i-2i00-1-5/2-1+5/21-5/21+5/21+5/21-5/2ζ43ζ5343ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ4ζ534ζ52    complex lifted from C4×D5
ρ18220-2-20-1-5/2-1+5/2-2i2i00-1+5/2-1-5/21+5/21-5/21-5/21+5/2ζ4ζ544ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5    complex lifted from C4×D5
ρ192-20-2i2i0-1-5/2-1+5/200001-5/21+5/2ζ86ζ5386ζ52ζ82ζ5482ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ83ζ5483ζ5ζ83ζ5383ζ5283ζ5383ζ52ζ87ζ5487ζ5ζ8ζ538ζ528ζ538ζ52ζ85ζ5485ζ5ζ8ζ548ζ5    complex faithful
ρ202-20-2i2i0-1-5/2-1+5/200001-5/21+5/2ζ86ζ5386ζ52ζ82ζ5482ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ87ζ5487ζ583ζ5383ζ52ζ83ζ5383ζ52ζ83ζ5483ζ58ζ538ζ52ζ8ζ538ζ52ζ8ζ548ζ5ζ85ζ5485ζ5    complex faithful
ρ212-20-2i2i0-1+5/2-1-5/200001+5/21-5/2ζ86ζ5486ζ5ζ82ζ5382ζ52ζ86ζ5386ζ52ζ82ζ5482ζ583ζ5383ζ52ζ83ζ5483ζ5ζ87ζ5487ζ5ζ83ζ5383ζ52ζ8ζ548ζ5ζ85ζ5485ζ5ζ8ζ538ζ528ζ538ζ52    complex faithful
ρ222-202i-2i0-1+5/2-1-5/200001+5/21-5/2ζ82ζ5482ζ5ζ86ζ5386ζ52ζ82ζ5382ζ52ζ86ζ5486ζ5ζ8ζ538ζ52ζ85ζ5485ζ5ζ8ζ548ζ58ζ538ζ52ζ87ζ5487ζ5ζ83ζ5483ζ583ζ5383ζ52ζ83ζ5383ζ52    complex faithful
ρ232-20-2i2i0-1+5/2-1-5/200001+5/21-5/2ζ86ζ5486ζ5ζ82ζ5382ζ52ζ86ζ5386ζ52ζ82ζ5482ζ5ζ83ζ5383ζ52ζ87ζ5487ζ5ζ83ζ5483ζ583ζ5383ζ52ζ85ζ5485ζ5ζ8ζ548ζ58ζ538ζ52ζ8ζ538ζ52    complex faithful
ρ242-202i-2i0-1-5/2-1+5/200001-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ5ζ82ζ5482ζ5ζ86ζ5386ζ52ζ85ζ5485ζ58ζ538ζ52ζ8ζ538ζ52ζ8ζ548ζ583ζ5383ζ52ζ83ζ5383ζ52ζ83ζ5483ζ5ζ87ζ5487ζ5    complex faithful
ρ252-202i-2i0-1+5/2-1-5/200001+5/21-5/2ζ82ζ5482ζ5ζ86ζ5386ζ52ζ82ζ5382ζ52ζ86ζ5486ζ58ζ538ζ52ζ8ζ548ζ5ζ85ζ5485ζ5ζ8ζ538ζ52ζ83ζ5483ζ5ζ87ζ5487ζ5ζ83ζ5383ζ5283ζ5383ζ52    complex faithful
ρ262-202i-2i0-1-5/2-1+5/200001-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ5ζ82ζ5482ζ5ζ86ζ5386ζ52ζ8ζ548ζ5ζ8ζ538ζ528ζ538ζ52ζ85ζ5485ζ5ζ83ζ5383ζ5283ζ5383ζ52ζ87ζ5487ζ5ζ83ζ5483ζ5    complex faithful

Smallest permutation representation of C8⋊D5
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)
(1 35 9 18 29)(2 36 10 19 30)(3 37 11 20 31)(4 38 12 21 32)(5 39 13 22 25)(6 40 14 23 26)(7 33 15 24 27)(8 34 16 17 28)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(10 14)(12 16)(17 38)(18 35)(19 40)(20 37)(21 34)(22 39)(23 36)(24 33)

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

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

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

C8⋊D5 is a maximal subgroup of
D20.3C4  D5×M4(2)  D20.2C4  D8⋊D5  D40⋊C2  SD16⋊D5  Q16⋊D5  C20.32D6  D30.5C4  C40⋊S3  C8⋊D25  C20.30D10  C20.31D10  C40⋊D5  D10.F5  Dic5.F5
C8⋊D5 is a maximal quotient of
C20.8Q8  C408C4  D101C8  C20.32D6  D30.5C4  C40⋊S3  C8⋊D25  C20.30D10  C20.31D10  C40⋊D5  D10.F5  Dic5.F5

Matrix representation of C8⋊D5 in GL2(𝔽29) generated by

111
1218
,
1218
1322
,
2226
167
G:=sub<GL(2,GF(29))| [11,12,1,18],[12,13,18,22],[22,16,26,7] >;

C8⋊D5 in GAP, Magma, Sage, TeX

C_8\rtimes D_5
% in TeX

G:=Group("C8:D5");
// GroupNames label

G:=SmallGroup(80,5);
// by ID

G=gap.SmallGroup(80,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,101,26,42,1604]);
// Polycyclic

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

Export

Subgroup lattice of C8⋊D5 in TeX
Character table of C8⋊D5 in TeX

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