metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D5, C22⋊Dic5, C10.11D4, C22.7D10, (C2×C10)⋊4C4, C5⋊3(C22⋊C4), C10.16(C2×C4), (C2×Dic5)⋊2C2, C2.3(C5⋊D4), C2.5(C2×Dic5), (C22×C10).2C2, (C2×C10).7C22, SmallGroup(80,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Character table of C23.D5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 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 | i | i | -i | -i | 1 | 1 | -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 | i | -i | -i | i | 1 | 1 | 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 | -i | -i | i | i | 1 | 1 | -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 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 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 D5 |
| 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 D5 |
| 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 D10 |
| 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 | symplectic lifted from Dic5, Schur index 2 |
| 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 | symplectic lifted from Dic5, Schur index 2 |
| 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 | symplectic lifted from Dic5, Schur index 2 |
| 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ53+ζ52 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | 1+√5/2 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ54-ζ5 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | -1+√5/2 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ53-ζ52 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 1-√5/2 | -1-√5/2 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ54+ζ5 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | -1+√5/2 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ53+ζ52 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | -1-√5/2 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ53-ζ52 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | 1+√5/2 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ54+ζ5 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | 1-√5/2 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ54-ζ5 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | 1-√5/2 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
►On 40 points
Generators in S40
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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)
(1 35 14 25)(2 34 15 24)(3 33 11 23)(4 32 12 22)(5 31 13 21)(6 38 16 28)(7 37 17 27)(8 36 18 26)(9 40 19 30)(10 39 20 29)
G:=sub<Sym(40)| (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,35,14,25)(2,34,15,24)(3,33,11,23)(4,32,12,22)(5,31,13,21)(6,38,16,28)(7,37,17,27)(8,36,18,26)(9,40,19,30)(10,39,20,29)>;
G:=Group( (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,35,14,25)(2,34,15,24)(3,33,11,23)(4,32,12,22)(5,31,13,21)(6,38,16,28)(7,37,17,27)(8,36,18,26)(9,40,19,30)(10,39,20,29) );
G=PermutationGroup([[(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,35,14,25),(2,34,15,24),(3,33,11,23),(4,32,12,22),(5,31,13,21),(6,38,16,28),(7,37,17,27),(8,36,18,26),(9,40,19,30),(10,39,20,29)]])
C23.D5 is a maximal subgroup of
C23.1D10 C23⋊Dic5 C23.11D10 Dic5.14D4 C23.D10 D5×C22⋊C4 D10.12D4 Dic5.5D4 C20.48D4 C23.21D10 C4×C5⋊D4 C23.23D10 D4×Dic5 C23.18D10 C20.17D4 C23⋊D10 C20⋊2D4 Dic5⋊D4 C24⋊2D5 D6⋊Dic5 C30.38D4 A4⋊Dic5 C23.D25 D10⋊Dic5 C102⋊11C4 D10.D10
C23.D5 is a maximal quotient of
C20.55D4 C10.10C42 D4⋊Dic5 C20.D4 C23⋊Dic5 Q8⋊Dic5 C20.10D4 D4⋊2Dic5 D6⋊Dic5 C30.38D4 C23.D25 D10⋊Dic5 C102⋊11C4 D10.D10
Matrix representation of C23.D5 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 10 |
| 32 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 37 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[1,0,0,0,37,0,0,0,10],[32,0,0,0,0,37,0,10,0] >;
C23.D5 in GAP, Magma, Sage, TeX
C_2^3.D_5
% in TeX
G:=Group("C2^3.D5"); // GroupNames label
G:=SmallGroup(80,19);
// by ID
G=gap.SmallGroup(80,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,1604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C23.D5 in TeX
Character table of C23.D5 in TeX