metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic5, C20.4C4, C5⋊4M4(2), C4.15D10, C22.Dic5, C20.15C22, C5⋊2C8⋊5C2, (C2×C4).2D5, (C2×C20).5C2, (C2×C10).5C4, C10.14(C2×C4), C2.3(C2×Dic5), SmallGroup(80,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic5
G = < a,b,c | a4=1, b10=a2, c2=b5, ab=ba, cac-1=a-1, cbc-1=b9 >
Character table of C4.Dic5
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 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 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | -2 | -1-√5/2 | -1+√5/2 | 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 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | 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 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | 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 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | 2 | 2 | -2 | -1+√5/2 | -1-√5/2 | 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 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 2 | -2 | -2 | -2 | -1+√5/2 | -1-√5/2 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | -2 | 2 | -1-√5/2 | -1+√5/2 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -2 | 2 | -1+√5/2 | -1-√5/2 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ16 | 2 | 2 | 2 | -2 | -2 | -2 | -1-√5/2 | -1+√5/2 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ17 | 2 | -2 | 0 | 2i | -2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2i | 0 | 0 | 0 | 2i | 2i | -2i | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | 0 | -2i | 2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2i | 0 | 0 | 0 | -2i | -2i | 2i | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | 0 | -2i | 2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | ζ4ζ53+ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ43ζ54-ζ43ζ5 | complex faithful |
| ρ20 | 2 | -2 | 0 | -2i | 2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | ζ4ζ54+ζ4ζ5 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | complex faithful |
| ρ21 | 2 | -2 | 0 | 2i | -2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | ζ43ζ53+ζ43ζ52 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ22 | 2 | -2 | 0 | -2i | 2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | ζ4ζ54+ζ4ζ5 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ23 | 2 | -2 | 0 | -2i | 2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | -ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ24 | 2 | -2 | 0 | 2i | -2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ4ζ53-ζ4ζ52 | complex faithful |
| ρ25 | 2 | -2 | 0 | 2i | -2i | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | ζ43ζ54+ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | -ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ26 | 2 | -2 | 0 | 2i | -2i | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | ζ43ζ53+ζ43ζ52 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | complex faithful |
►On 40 points
Generators in S40
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(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 31 6 36 11 21 16 26)(2 40 7 25 12 30 17 35)(3 29 8 34 13 39 18 24)(4 38 9 23 14 28 19 33)(5 27 10 32 15 37 20 22)
G:=sub<Sym(40)| (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (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), (1,31,6,36,11,21,16,26)(2,40,7,25,12,30,17,35)(3,29,8,34,13,39,18,24)(4,38,9,23,14,28,19,33)(5,27,10,32,15,37,20,22)>;
G:=Group( (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (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), (1,31,6,36,11,21,16,26)(2,40,7,25,12,30,17,35)(3,29,8,34,13,39,18,24)(4,38,9,23,14,28,19,33)(5,27,10,32,15,37,20,22) );
G=PermutationGroup([[(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30)], [(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)], [(1,31,6,36,11,21,16,26),(2,40,7,25,12,30,17,35),(3,29,8,34,13,39,18,24),(4,38,9,23,14,28,19,33),(5,27,10,32,15,37,20,22)]])
C4.Dic5 is a maximal subgroup of
D20⋊4C4 C40.6C4 C20.53D4 C20.46D4 C4.12D20 C20.D4 C20.10D4 D4⋊2Dic5 D20.3C4 D5×M4(2) D4.D10 C20.C23 D4.Dic5 D4⋊D10 D4.9D10 D6.Dic5 C60.7C4 C4.Dic25 C20.30D10 C20.59D10 C20.12F5 C102.C4
C4.Dic5 is a maximal quotient of
C42.D5 C20⋊3C8 C20.55D4 D6.Dic5 C60.7C4 C4.Dic25 C20.30D10 C20.59D10 C20.12F5 C102.C4
Matrix representation of C4.Dic5 ►in GL2(𝔽41) generated by
| 32 | 0 |
| 0 | 9 |
| 36 | 0 |
| 0 | 33 |
| 0 | 1 |
| 32 | 0 |
G:=sub<GL(2,GF(41))| [32,0,0,9],[36,0,0,33],[0,32,1,0] >;
C4.Dic5 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_5 % in TeX
G:=Group("C4.Dic5"); // GroupNames label
G:=SmallGroup(80,10);
// by ID
G=gap.SmallGroup(80,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^10=a^2,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^9>;
// generators/relations
Export
Subgroup lattice of C4.Dic5 in TeX
Character table of C4.Dic5 in TeX