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G = C40⋊C2  order 80 = 24·5

2nd semidirect product of C40 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C8⋊2D5, C40⋊2C2, C5⋊1SD16, C2.3D20, C10.1D4, C4.8D10, D20.1C2, Dic10⋊1C2, C20.8C22, SmallGroup(80,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C40⋊C2
Chief series
C1 — C5 — C10 — C20 — D20 — C40⋊C2
Lower central
C5 — C10 — C20 — C40⋊C2
Upper central
C1 — C2 — C4 — C8

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

C1
C2
20C2
C5
C4
10C22
10C4
C10
4D5
C8
5Q8
5D4
C20
2Dic5
2D10
5SD16
D20
Dic10
C40
C40⋊C2

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-√5/2-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ11220-20-1-√5/2-1+√5/200-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ12220-20-1+√5/2-1-√5/200-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ13220-20-1-√5/2-1+√5/200-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    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-√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5ζ87ζ5+ζ85ζ54ζ87ζ54+ζ85ζ5ζ83ζ54+ζ8ζ5ζ83ζ53+ζ8ζ52ζ83ζ52+ζ8ζ53ζ83ζ5+ζ8ζ54ζ87ζ53+ζ85ζ52ζ87ζ52+ζ85ζ53    complex faithful
ρ172-2000-1+√5/2-1-√5/2-√-2√-21+√5/21-√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5ζ87ζ54+ζ85ζ5ζ87ζ5+ζ85ζ54ζ83ζ5+ζ8ζ54ζ83ζ52+ζ8ζ53ζ83ζ53+ζ8ζ52ζ83ζ54+ζ8ζ5ζ87ζ52+ζ85ζ53ζ87ζ53+ζ85ζ52    complex faithful
ρ182-2000-1-√5/2-1+√5/2√-2-√-21-√5/21+√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52ζ83ζ53+ζ8ζ52ζ83ζ52+ζ8ζ53ζ87ζ52+ζ85ζ53ζ87ζ54+ζ85ζ5ζ87ζ5+ζ85ζ54ζ87ζ53+ζ85ζ52ζ83ζ54+ζ8ζ5ζ83ζ5+ζ8ζ54    complex faithful
ρ192-2000-1-√5/2-1+√5/2-√-2√-21-√5/21+√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52ζ87ζ52+ζ85ζ53ζ87ζ53+ζ85ζ52ζ83ζ53+ζ8ζ52ζ83ζ5+ζ8ζ54ζ83ζ54+ζ8ζ5ζ83ζ52+ζ8ζ53ζ87ζ5+ζ85ζ54ζ87ζ54+ζ85ζ5    complex faithful
ρ202-2000-1+√5/2-1-√5/2√-2-√-21+√5/21-√5/2ζ86ζ54-ζ86ζ5-ζ82ζ53+ζ82ζ52ζ82ζ53-ζ82ζ52-ζ86ζ54+ζ86ζ5ζ83ζ5+ζ8ζ54ζ83ζ54+ζ8ζ5ζ87ζ54+ζ85ζ5ζ87ζ53+ζ85ζ52ζ87ζ52+ζ85ζ53ζ87ζ5+ζ85ζ54ζ83ζ53+ζ8ζ52ζ83ζ52+ζ8ζ53    complex faithful
ρ212-2000-1-√5/2-1+√5/2-√-2√-21-√5/21+√5/2ζ82ζ53-ζ82ζ52ζ86ζ54-ζ86ζ5-ζ86ζ54+ζ86ζ5-ζ82ζ53+ζ82ζ52ζ87ζ53+ζ85ζ52ζ87ζ52+ζ85ζ53ζ83ζ52+ζ8ζ53ζ83ζ54+ζ8ζ5ζ83ζ5+ζ8ζ54ζ83ζ53+ζ8ζ52ζ87ζ54+ζ85ζ5ζ87ζ5+ζ85ζ54    complex faithful
ρ222-2000-1-√5/2-1+√5/2√-2-√-21-√5/21+√5/2-ζ82ζ53+ζ82ζ52-ζ86ζ54+ζ86ζ5ζ86ζ54-ζ86ζ5ζ82ζ53-ζ82ζ52ζ83ζ52+ζ8ζ53ζ83ζ53+ζ8ζ52ζ87ζ53+ζ85ζ52ζ87ζ5+ζ85ζ54ζ87ζ54+ζ85ζ5ζ87ζ52+ζ85ζ53ζ83ζ5+ζ8ζ54ζ83ζ54+ζ8ζ5    complex faithful
ρ232-2000-1+√5/2-1-√5/2√-2-√-21+√5/21-√5/2-ζ86ζ54+ζ86ζ5ζ82ζ53-ζ82ζ52-ζ82ζ53+ζ82ζ52ζ86ζ54-ζ86ζ5ζ83ζ54+ζ8ζ5ζ83ζ5+ζ8ζ54ζ87ζ5+ζ85ζ54ζ87ζ52+ζ85ζ53ζ87ζ53+ζ85ζ52ζ87ζ54+ζ85ζ5ζ83ζ52+ζ8ζ53ζ83ζ53+ζ8ζ52    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
 D40⋊7C2  C8⋊D10  C8.D10  D8⋊D5  D5×SD16  SD16⋊3D5  Q16⋊D5  C6.D20  C15⋊SD16  C24⋊D5  C200⋊C2  C52⋊3SD16  C52⋊4SD16  C40⋊2D5
C40⋊C2 is a maximal quotient of
 C20.44D4  C40⋊6C4  D20⋊5C4  C6.D20  C15⋊SD16  C24⋊D5  C200⋊C2  C52⋊3SD16  C52⋊4SD16  C40⋊2D5

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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