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G = C2×C5⋊D4  order 80 = 24·5

Direct product of C2 and C5⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C5⋊D4, C23⋊D5, C10⋊2D4, C22⋊2D10, D10⋊3C22, C10.10C23, Dic5⋊2C22, C5⋊3(C2×D4), (C2×C10)⋊3C22, (C22×C10)⋊2C2, (C2×Dic5)⋊4C2, (C22×D5)⋊3C2, C2.10(C22×D5), SmallGroup(80,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×C5⋊D4
Chief series
C1 — C5 — C10 — D10 — C22×D5 — C2×C5⋊D4
Lower central
C5 — C10 — C2×C5⋊D4
Upper central
C1 — C22 — C23

Generators and relations for C2×C5⋊D4
 G = < a,b,c,d | a2=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 146 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C10, C2×D4, Dic5, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×C5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C5⋊D4, C22×D5, C2×C5⋊D4

Character table of C2×C5⋊D4

 class 12A2B2C2D2E2F2G4A4B5A5B10A10B10C10D10E10F10G10H10I10J10K10L10M10N
 size 111122101010102222222222222222
ρ111111111111111111111111111    trivial
ρ21-1-111-11-1-11111-11-1-11-1-1-1-1-1111    linear of order 2
ρ31111-1-111-1-111-111111-1-1-1-11-1-1-1    linear of order 2
ρ41-1-11-111-11-111-1-11-1-111111-1-1-1-1    linear of order 2
ρ51111-1-1-1-11111-111111-1-1-1-11-1-1-1    linear of order 2
ρ61-1-11-11-11-1111-1-11-1-111111-1-1-1-1    linear of order 2
ρ7111111-1-1-1-11111111111111111    linear of order 2
ρ81-1-111-1-111-1111-11-1-11-1-1-1-1-1111    linear of order 2
ρ92-22-20000002202-2-2-2-200002000    orthogonal lifted from D4
ρ1022-2-2000000220-2-222-20000-2000    orthogonal lifted from D4
ρ112-2-22-220000-1-√5/2-1+√5/21-√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ122-2-222-20000-1-√5/2-1+√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D10
ρ132222-2-20000-1-√5/2-1+√5/21-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/2-1+√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ142222220000-1+√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-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ152-2-222-20000-1+√5/2-1-√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D10
ρ162222220000-1-√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+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ172222-2-20000-1+√5/2-1-√5/21+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/2-1-√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ182-2-22-220000-1+√5/2-1-√5/21+√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ192-22-2000000-1-√5/2-1+√5/2-ζ54+ζ5-1-√5/21-√5/21-√5/21+√5/21+√5/2ζ54-ζ5ζ53-ζ52-ζ53+ζ52-ζ54+ζ5-1+√5/2ζ54-ζ5ζ53-ζ52-ζ53+ζ52    complex lifted from C5⋊D4
ρ2022-2-2000000-1-√5/2-1+√5/2ζ54-ζ51+√5/21-√5/2-1+√5/2-1-√5/21+√5/2ζ54-ζ5ζ53-ζ52-ζ53+ζ52-ζ54+ζ51-√5/2-ζ54+ζ5-ζ53+ζ52ζ53-ζ52    complex lifted from C5⋊D4
ρ2122-2-2000000-1+√5/2-1-√5/2-ζ53+ζ521-√5/21+√5/2-1-√5/2-1+√5/21-√5/2-ζ53+ζ52ζ54-ζ5-ζ54+ζ5ζ53-ζ521+√5/2ζ53-ζ52-ζ54+ζ5ζ54-ζ5    complex lifted from C5⋊D4
ρ2222-2-2000000-1-√5/2-1+√5/2-ζ54+ζ51+√5/21-√5/2-1+√5/2-1-√5/21+√5/2-ζ54+ζ5-ζ53+ζ52ζ53-ζ52ζ54-ζ51-√5/2ζ54-ζ5ζ53-ζ52-ζ53+ζ52    complex lifted from C5⋊D4
ρ232-22-2000000-1-√5/2-1+√5/2ζ54-ζ5-1-√5/21-√5/21-√5/21+√5/21+√5/2-ζ54+ζ5-ζ53+ζ52ζ53-ζ52ζ54-ζ5-1+√5/2-ζ54+ζ5-ζ53+ζ52ζ53-ζ52    complex lifted from C5⋊D4
ρ2422-2-2000000-1+√5/2-1-√5/2ζ53-ζ521-√5/21+√5/2-1-√5/2-1+√5/21-√5/2ζ53-ζ52-ζ54+ζ5ζ54-ζ5-ζ53+ζ521+√5/2-ζ53+ζ52ζ54-ζ5-ζ54+ζ5    complex lifted from C5⋊D4
ρ252-22-2000000-1+√5/2-1-√5/2-ζ53+ζ52-1+√5/21+√5/21+√5/21-√5/21-√5/2ζ53-ζ52-ζ54+ζ5ζ54-ζ5-ζ53+ζ52-1-√5/2ζ53-ζ52-ζ54+ζ5ζ54-ζ5    complex lifted from C5⋊D4
ρ262-22-2000000-1+√5/2-1-√5/2ζ53-ζ52-1+√5/21+√5/21+√5/21-√5/21-√5/2-ζ53+ζ52ζ54-ζ5-ζ54+ζ5ζ53-ζ52-1-√5/2-ζ53+ζ52ζ54-ζ5-ζ54+ζ5    complex lifted from C5⋊D4

Smallest permutation representation of C2×C5⋊D4
►On 40 points
Generators in S40
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)

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

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

C2×C5⋊D4 is a maximal subgroup of
 C23.1D10  C23⋊F5  C23.F5  Dic5⋊4D4  C22⋊D20  D10.12D4  D10⋊D4  Dic5.5D4  C22.D20  C23.23D10  C20⋊7D4  C23⋊D10  C20⋊2D4  Dic5⋊D4  C20⋊D4  C24⋊2D5  C2×D4×D5  D4⋊6D10
C2×C5⋊D4 is a maximal quotient of
 C20.48D4  C23.23D10  C20⋊7D4  D4.D10  C23.18D10  C20.17D4  C23⋊D10  C20⋊2D4  Dic5⋊D4  C20⋊D4  C20.C23  Dic5⋊Q8  D10⋊3Q8  C20.23D4  D4⋊D10  D4.8D10  D4.9D10  C24⋊2D5

Matrix representation of C2×C5⋊D4 ►in GL3(𝔽41) generated by

4000
0400
0040
,
100
0640
010
,
4000
01835
02023
,
4000
010
0640
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,6,1,0,40,0],[40,0,0,0,18,20,0,35,23],[40,0,0,0,1,6,0,0,40] >;
 

C2×C5⋊D4 in GAP, Magma, Sage, TeX 

C_2\times C_5\rtimes D_4
 
% in TeX
 
G:=Group("C2xC5:D4");
 
// GroupNames label
 
G:=SmallGroup(80,44);
 
// by ID
 
G=gap.SmallGroup(80,44);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-5,182,1604]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C2×C5⋊D4 in TeX

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