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G = C2xC5:D4  order 80 = 24·5

Direct product of C2 and C5:D4

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

Aliases: C2xC5:D4, C23:D5, C10:2D4, C22:2D10, D10:3C22, C10.10C23, Dic5:2C22, C5:3(C2xD4), (C2xC10):3C22, (C22xC10):2C2, (C2xDic5):4C2, (C22xD5):3C2, C2.10(C22xD5), SmallGroup(80,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2xC5:D4
Chief series
C1 — C5 — C10 — D10 — C22xD5 — C2xC5:D4
Lower central
C5 — C10 — C2xC5:D4
Upper central
C1 — C22 — C23

Generators and relations for C2xC5: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, C2xC4, D4, C23, C23, D5, C10, C10, C10, C2xD4, Dic5, D10, D10, C2xC10, C2xC10, C2xC10, C2xDic5, C5:D4, C22xD5, C22xC10, C2xC5:D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C5:D4, C22xD5, C2xC5:D4

Character table of C2xC5: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 C2xC5: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)]])
 

C2xC5: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  C2xD4xD5  D4:6D10
C2xC5: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 C2xC5: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] >;
 

C2xC5: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 C2xC5:D4 in TeX

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