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
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 | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ8 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -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 |
ρ15 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ16 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -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 |
ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -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 D10 |
ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ54+ζ5 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | -1+√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | complex lifted from C5⋊D4 |
ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ54-ζ5 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | complex lifted from C5⋊D4 |
ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ53+ζ52 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | complex lifted from C5⋊D4 |
ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ54+ζ5 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | complex lifted from C5⋊D4 |
ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ54-ζ5 | -1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -1+√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | complex lifted from C5⋊D4 |
ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ53-ζ52 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | complex lifted from C5⋊D4 |
ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ53+ζ52 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | -1-√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | complex lifted from C5⋊D4 |
ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ53-ζ52 | -1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | -1-√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | complex lifted from C5⋊D4 |
(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
40 | 0 | 0 |
0 | 40 | 0 |
0 | 0 | 40 |
1 | 0 | 0 |
0 | 6 | 40 |
0 | 1 | 0 |
40 | 0 | 0 |
0 | 18 | 35 |
0 | 20 | 23 |
40 | 0 | 0 |
0 | 1 | 0 |
0 | 6 | 40 |
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
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