direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C4×C11⋊C5, C44⋊C5, C11⋊2C20, C22.2C10, C2.(C2×C11⋊C5), (C2×C11⋊C5).2C2, SmallGroup(220,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C4×C11⋊C5 |
Generators and relations for C4×C11⋊C5
G = < a,b,c | a4=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C4×C11⋊C5
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | 11B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 22A | 22B | 44A | 44B | 44C | 44D | |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 5 | 5 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 5 | 5 | 5 | 5 | 5 | 5 | |
| 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 | 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 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | -i | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | i | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ5 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | 1 | ζ52 | ζ53 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ6 | 1 | 1 | -1 | -1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | 1 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ7 | 1 | 1 | -1 | -1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | 1 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ8 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | 1 | ζ5 | ζ54 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ9 | 1 | 1 | -1 | -1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | 1 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ10 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | 1 | ζ53 | ζ52 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ11 | 1 | 1 | -1 | -1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | 1 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ12 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | 1 | ζ54 | ζ5 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ13 | 1 | -1 | -i | i | ζ54 | ζ5 | ζ52 | ζ53 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | 1 | 1 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ54 | -1 | -1 | -i | -i | i | i | linear of order 20 |
| ρ14 | 1 | -1 | -i | i | ζ5 | ζ54 | ζ53 | ζ52 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | 1 | 1 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ5 | -1 | -1 | -i | -i | i | i | linear of order 20 |
| ρ15 | 1 | -1 | i | -i | ζ52 | ζ53 | ζ5 | ζ54 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | 1 | 1 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ52 | -1 | -1 | i | i | -i | -i | linear of order 20 |
| ρ16 | 1 | -1 | i | -i | ζ53 | ζ52 | ζ54 | ζ5 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | 1 | 1 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ53 | -1 | -1 | i | i | -i | -i | linear of order 20 |
| ρ17 | 1 | -1 | i | -i | ζ5 | ζ54 | ζ53 | ζ52 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | 1 | 1 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ5 | -1 | -1 | i | i | -i | -i | linear of order 20 |
| ρ18 | 1 | -1 | -i | i | ζ52 | ζ53 | ζ5 | ζ54 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | 1 | 1 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ52 | -1 | -1 | -i | -i | i | i | linear of order 20 |
| ρ19 | 1 | -1 | i | -i | ζ54 | ζ5 | ζ52 | ζ53 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | 1 | 1 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ54 | -1 | -1 | i | i | -i | -i | linear of order 20 |
| ρ20 | 1 | -1 | -i | i | ζ53 | ζ52 | ζ54 | ζ5 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | 1 | 1 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ53 | -1 | -1 | -i | -i | i | i | linear of order 20 |
| ρ21 | 5 | 5 | -5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | 1-√-11/2 | 1+√-11/2 | 1+√-11/2 | 1-√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ22 | 5 | 5 | -5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | 1+√-11/2 | 1-√-11/2 | 1-√-11/2 | 1+√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ23 | 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | -1-√-11/2 | -1+√-11/2 | -1+√-11/2 | -1-√-11/2 | complex lifted from C11⋊C5 |
| ρ24 | 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | -1+√-11/2 | -1-√-11/2 | -1-√-11/2 | -1+√-11/2 | complex lifted from C11⋊C5 |
| ρ25 | 5 | -5 | 5i | -5i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-11/2 | 1+√-11/2 | ζ4ζ1110+ζ4ζ118+ζ4ζ117+ζ4ζ116+ζ4ζ112 | ζ4ζ119+ζ4ζ115+ζ4ζ114+ζ4ζ113+ζ4ζ11 | ζ43ζ119+ζ43ζ115+ζ43ζ114+ζ43ζ113+ζ43ζ11 | ζ43ζ1110+ζ43ζ118+ζ43ζ117+ζ43ζ116+ζ43ζ112 | complex faithful |
| ρ26 | 5 | -5 | 5i | -5i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-11/2 | 1-√-11/2 | ζ4ζ119+ζ4ζ115+ζ4ζ114+ζ4ζ113+ζ4ζ11 | ζ4ζ1110+ζ4ζ118+ζ4ζ117+ζ4ζ116+ζ4ζ112 | ζ43ζ1110+ζ43ζ118+ζ43ζ117+ζ43ζ116+ζ43ζ112 | ζ43ζ119+ζ43ζ115+ζ43ζ114+ζ43ζ113+ζ43ζ11 | complex faithful |
| ρ27 | 5 | -5 | -5i | 5i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-11/2 | -1-√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-11/2 | 1+√-11/2 | ζ43ζ1110+ζ43ζ118+ζ43ζ117+ζ43ζ116+ζ43ζ112 | ζ43ζ119+ζ43ζ115+ζ43ζ114+ζ43ζ113+ζ43ζ11 | ζ4ζ119+ζ4ζ115+ζ4ζ114+ζ4ζ113+ζ4ζ11 | ζ4ζ1110+ζ4ζ118+ζ4ζ117+ζ4ζ116+ζ4ζ112 | complex faithful |
| ρ28 | 5 | -5 | -5i | 5i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-11/2 | -1+√-11/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-11/2 | 1-√-11/2 | ζ43ζ119+ζ43ζ115+ζ43ζ114+ζ43ζ113+ζ43ζ11 | ζ43ζ1110+ζ43ζ118+ζ43ζ117+ζ43ζ116+ζ43ζ112 | ζ4ζ1110+ζ4ζ118+ζ4ζ117+ζ4ζ116+ζ4ζ112 | ζ4ζ119+ζ4ζ115+ζ4ζ114+ζ4ζ113+ζ4ζ11 | complex faithful |
►On 44 points
Generators in S44
(1 34 12 23)(2 35 13 24)(3 36 14 25)(4 37 15 26)(5 38 16 27)(6 39 17 28)(7 40 18 29)(8 41 19 30)(9 42 20 31)(10 43 21 32)(11 44 22 33)
(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 41 42 43 44)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)
G:=sub<Sym(44)| (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33), (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)(34,35,36,37,38,39,40,41,42,43,44), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)>;
G:=Group( (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33), (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)(34,35,36,37,38,39,40,41,42,43,44), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40) );
G=PermutationGroup([[(1,34,12,23),(2,35,13,24),(3,36,14,25),(4,37,15,26),(5,38,16,27),(6,39,17,28),(7,40,18,29),(8,41,19,30),(9,42,20,31),(10,43,21,32),(11,44,22,33)], [(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),(34,35,36,37,38,39,40,41,42,43,44)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40)]])
C4×C11⋊C5 is a maximal subgroup of
C11⋊C40 C4.F11 D44⋊C5
Matrix representation of C4×C11⋊C5 ►in GL6(𝔽661)
| 555 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 43 | 2 | 615 | 44 | 1 |
| 0 | 44 | 2 | 615 | 44 | 1 |
| 0 | 43 | 3 | 615 | 44 | 1 |
| 0 | 43 | 2 | 616 | 44 | 1 |
| 0 | 43 | 2 | 615 | 45 | 1 |
| 247 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 45 | 618 | 659 | 46 | 617 |
| 0 | 46 | 617 | 43 | 2 | 616 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(661))| [555,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,43,44,43,43,43,0,2,2,3,2,2,0,615,615,615,616,615,0,44,44,44,44,45,0,1,1,1,1,1],[247,0,0,0,0,0,0,0,45,46,1,0,0,0,618,617,0,0,0,1,659,43,0,0,0,0,46,2,0,1,0,0,617,616,0,0] >;
C4×C11⋊C5 in GAP, Magma, Sage, TeX
C_4\times C_{11}\rtimes C_5 % in TeX
G:=Group("C4xC11:C5"); // GroupNames label
G:=SmallGroup(220,2);
// by ID
G=gap.SmallGroup(220,2);
# by ID
G:=PCGroup([4,-2,-5,-2,-11,40,647]);
// Polycyclic
G:=Group<a,b,c|a^4=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C4×C11⋊C5 in TeX
Character table of C4×C11⋊C5 in TeX