direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C22×C11⋊C5, C22⋊2C10, (C2×C22)⋊C5, C11⋊2(C2×C10), SmallGroup(220,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C22×C11⋊C5 |
Generators and relations for C22×C11⋊C5
G = < a,b,c,d | a2=b2=c11=d5=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Character table of C22×C11⋊C5
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 11A | 11B | 22A | 22B | 22C | 22D | 22E | 22F | |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 5 | 5 | 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 | -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 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ6 | 1 | 1 | -1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ54 | -ζ53 | -ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ52 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 10 |
| ρ7 | 1 | 1 | -1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ53 | -ζ5 | -ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ54 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 10 |
| ρ8 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ52 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ9 | 1 | 1 | -1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ52 | -ζ54 | -ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ5 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 10 |
| ρ10 | 1 | -1 | -1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | ζ52 | ζ54 | ζ5 | -ζ54 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ11 | 1 | -1 | 1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ53 | ζ5 | ζ53 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ52 | -ζ52 | -ζ54 | -ζ5 | ζ54 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 10 |
| ρ12 | 1 | -1 | -1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | ζ54 | ζ53 | ζ52 | -ζ53 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ13 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ53 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ14 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ54 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ15 | 1 | -1 | 1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ5 | ζ52 | ζ5 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ54 | -ζ54 | -ζ53 | -ζ52 | ζ53 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 10 |
| ρ16 | 1 | -1 | 1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ54 | ζ53 | ζ54 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ5 | -ζ5 | -ζ52 | -ζ53 | ζ52 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 10 |
| ρ17 | 1 | 1 | -1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ5 | -ζ52 | -ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ53 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 10 |
| ρ18 | 1 | -1 | -1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | ζ5 | ζ52 | ζ53 | -ζ52 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ19 | 1 | -1 | -1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | ζ53 | ζ5 | ζ54 | -ζ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 10 |
| ρ20 | 1 | -1 | 1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ52 | ζ54 | ζ52 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ53 | -ζ53 | -ζ5 | -ζ54 | ζ5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 10 |
| ρ21 | 5 | -5 | -5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 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 | 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 | -1+√-11/2 | -1-√-11/2 | complex lifted from C11⋊C5 |
| ρ23 | 5 | -5 | -5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 1+√-11/2 | 1-√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ24 | 5 | 5 | -5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 1-√-11/2 | -1-√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ25 | 5 | -5 | 5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -1+√-11/2 | 1+√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ26 | 5 | -5 | 5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -1-√-11/2 | 1-√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ27 | 5 | 5 | -5 | -5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 1+√-11/2 | -1+√-11/2 | complex lifted from C2×C11⋊C5 |
| ρ28 | 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -1-√-11/2 | -1+√-11/2 | complex lifted from C11⋊C5 |
►On 44 points
Generators in S44
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)
(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)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44)], [(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)]])
C22×C11⋊C5 is a maximal subgroup of
C22⋊F11
Matrix representation of C22×C11⋊C5 ►in GL6(𝔽331)
| 330 | 0 | 0 | 0 | 0 | 0 |
| 0 | 330 | 0 | 0 | 0 | 0 |
| 0 | 0 | 330 | 0 | 0 | 0 |
| 0 | 0 | 0 | 330 | 0 | 0 |
| 0 | 0 | 0 | 0 | 330 | 0 |
| 0 | 0 | 0 | 0 | 0 | 330 |
| 330 | 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 | 103 | 2 | 225 | 104 | 1 |
| 0 | 104 | 2 | 225 | 104 | 1 |
| 0 | 103 | 3 | 225 | 104 | 1 |
| 0 | 103 | 2 | 226 | 104 | 1 |
| 0 | 103 | 2 | 225 | 105 | 1 |
| 323 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 105 | 228 | 329 | 106 | 227 |
| 0 | 106 | 227 | 103 | 2 | 226 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(331))| [330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330],[330,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,103,104,103,103,103,0,2,2,3,2,2,0,225,225,225,226,225,0,104,104,104,104,105,0,1,1,1,1,1],[323,0,0,0,0,0,0,0,105,106,1,0,0,0,228,227,0,0,0,1,329,103,0,0,0,0,106,2,0,1,0,0,227,226,0,0] >;
C22×C11⋊C5 in GAP, Magma, Sage, TeX
C_2^2\times C_{11}\rtimes C_5 % in TeX
G:=Group("C2^2xC11:C5"); // GroupNames label
G:=SmallGroup(220,8);
// by ID
G=gap.SmallGroup(220,8);
# by ID
G:=PCGroup([4,-2,-2,-5,-11,331]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^11=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C22×C11⋊C5 in TeX
Character table of C22×C11⋊C5 in TeX