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G = C40⋊C2order 80 = 24·5

2nd semidirect product of C40 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D5, C402C2, C51SD16, C2.3D20, C10.1D4, C4.8D10, D20.1C2, Dic101C2, C20.8C22, SmallGroup(80,6)

Series: Derived Chief Lower central Upper central

C1C20 — C40⋊C2
C1C5C10C20D20 — C40⋊C2
C5C10C20 — C40⋊C2
C1C2C4C8

Generators and relations for C40⋊C2
 G = < a,b | a40=b2=1, bab=a19 >

20C2
10C22
10C4
4D5
5Q8
5D4
2Dic5
2D10
5SD16

Character table of C40⋊C2

 class 12A2B4A4B5A5B8A8B10A10B20A20B20C20D40A40B40C40D40E40F40G40H
 size 1120220222222222222222222
ρ111111111111111111111111    trivial
ρ211-11-1111111111111111111    linear of order 2
ρ31111-111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-11111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5220-20220022-2-2-2-200000000    orthogonal lifted from D4
ρ622020-1+5/2-1-5/222-1-5/2-1+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
ρ722020-1-5/2-1+5/2-2-2-1+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
ρ822020-1-5/2-1+5/222-1+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
ρ922020-1+5/2-1-5/2-2-2-1-5/2-1+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
ρ10220-20-1+5/2-1-5/200-1-5/2-1+5/21-5/21+5/21+5/21-5/243ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    orthogonal lifted from D20
ρ11220-20-1-5/2-1+5/200-1+5/2-1-5/21+5/21-5/21-5/21+5/24ζ534ζ52ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    orthogonal lifted from D20
ρ12220-20-1+5/2-1-5/200-1-5/2-1+5/21-5/21+5/21+5/21-5/2ζ43ζ5443ζ543ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ13220-20-1-5/2-1+5/200-1+5/2-1-5/21+5/21-5/21-5/21+5/2ζ4ζ534ζ524ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    orthogonal lifted from D20
ρ142-200022--2-2-2-20000--2--2-2-2-2-2--2--2    complex lifted from SD16
ρ152-200022-2--2-2-20000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ162-2000-1+5/2-1-5/2--2-21+5/21-5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ5ζ87ζ585ζ54ζ87ζ5485ζ5ζ83ζ548ζ5ζ83ζ538ζ52ζ83ζ528ζ53ζ83ζ58ζ54ζ87ζ5385ζ52ζ87ζ5285ζ53    complex faithful
ρ172-2000-1+5/2-1-5/2--2-21+5/21-5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ5ζ87ζ5485ζ5ζ87ζ585ζ54ζ83ζ58ζ54ζ83ζ528ζ53ζ83ζ538ζ52ζ83ζ548ζ5ζ87ζ5285ζ53ζ87ζ5385ζ52    complex faithful
ρ182-2000-1-5/2-1+5/2-2--21-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ52ζ83ζ538ζ52ζ83ζ528ζ53ζ87ζ5285ζ53ζ87ζ5485ζ5ζ87ζ585ζ54ζ87ζ5385ζ52ζ83ζ548ζ5ζ83ζ58ζ54    complex faithful
ρ192-2000-1-5/2-1+5/2--2-21-5/21+5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ87ζ5285ζ53ζ87ζ5385ζ52ζ83ζ538ζ52ζ83ζ58ζ54ζ83ζ548ζ5ζ83ζ528ζ53ζ87ζ585ζ54ζ87ζ5485ζ5    complex faithful
ρ202-2000-1+5/2-1-5/2-2--21+5/21-5/2ζ86ζ5486ζ582ζ5382ζ52ζ82ζ5382ζ5286ζ5486ζ5ζ83ζ58ζ54ζ83ζ548ζ5ζ87ζ5485ζ5ζ87ζ5385ζ52ζ87ζ5285ζ53ζ87ζ585ζ54ζ83ζ538ζ52ζ83ζ528ζ53    complex faithful
ρ212-2000-1-5/2-1+5/2--2-21-5/21+5/2ζ82ζ5382ζ52ζ86ζ5486ζ586ζ5486ζ582ζ5382ζ52ζ87ζ5385ζ52ζ87ζ5285ζ53ζ83ζ528ζ53ζ83ζ548ζ5ζ83ζ58ζ54ζ83ζ538ζ52ζ87ζ5485ζ5ζ87ζ585ζ54    complex faithful
ρ222-2000-1-5/2-1+5/2-2--21-5/21+5/282ζ5382ζ5286ζ5486ζ5ζ86ζ5486ζ5ζ82ζ5382ζ52ζ83ζ528ζ53ζ83ζ538ζ52ζ87ζ5385ζ52ζ87ζ585ζ54ζ87ζ5485ζ5ζ87ζ5285ζ53ζ83ζ58ζ54ζ83ζ548ζ5    complex faithful
ρ232-2000-1+5/2-1-5/2-2--21+5/21-5/286ζ5486ζ5ζ82ζ5382ζ5282ζ5382ζ52ζ86ζ5486ζ5ζ83ζ548ζ5ζ83ζ58ζ54ζ87ζ585ζ54ζ87ζ5285ζ53ζ87ζ5385ζ52ζ87ζ5485ζ5ζ83ζ528ζ53ζ83ζ538ζ52    complex faithful

Smallest permutation representation of C40⋊C2
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 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)

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

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

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

C40⋊C2 is a maximal subgroup of
D407C2  C8⋊D10  C8.D10  D8⋊D5  D5×SD16  SD163D5  Q16⋊D5  C6.D20  C15⋊SD16  C24⋊D5  C200⋊C2  C523SD16  C524SD16  C402D5
C40⋊C2 is a maximal quotient of
C20.44D4  C406C4  D205C4  C6.D20  C15⋊SD16  C24⋊D5  C200⋊C2  C523SD16  C524SD16  C402D5

Matrix representation of C40⋊C2 in GL2(𝔽19) generated by

1410
103
,
141
145
G:=sub<GL(2,GF(19))| [14,10,10,3],[14,14,1,5] >;

C40⋊C2 in GAP, Magma, Sage, TeX

C_{40}\rtimes C_2
% in TeX

G:=Group("C40:C2");
// GroupNames label

G:=SmallGroup(80,6);
// by ID

G=gap.SmallGroup(80,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,61,26,182,42,1604]);
// Polycyclic

G:=Group<a,b|a^40=b^2=1,b*a*b=a^19>;
// generators/relations

Export

Subgroup lattice of C40⋊C2 in TeX
Character table of C40⋊C2 in TeX

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