direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D11, C11⋊C23, C22⋊C22, (C2×C22)⋊3C2, SmallGroup(88,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C22×D11 |
Generators and relations for C22×D11
G = < a,b,c,d | a2=b2=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of C22×D11
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | 22F | 22G | 22H | 22I | 22J | 22K | 22L | 22M | 22N | 22O | |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | 2 | 2 | 2 | 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 | 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 | -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 | -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 | 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 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ118-ζ113 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ1110-ζ11 | ζ117+ζ114 | -ζ117-ζ114 | orthogonal lifted from D22 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | ζ1110+ζ11 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | ζ117+ζ114 | ζ119+ζ112 | -ζ118-ζ113 | ζ118+ζ113 | orthogonal lifted from D22 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ117-ζ114 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ116-ζ115 | ζ119+ζ112 | -ζ119-ζ112 | orthogonal lifted from D22 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ118-ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ119-ζ112 | orthogonal lifted from D22 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ1110-ζ11 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ117-ζ114 | ζ116+ζ115 | -ζ116-ζ115 | orthogonal lifted from D22 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ119+ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ1110-ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ118-ζ113 | orthogonal lifted from D22 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | ζ116+ζ115 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | ζ119+ζ112 | ζ1110+ζ11 | -ζ117-ζ114 | ζ117+ζ114 | orthogonal lifted from D22 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ117-ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ1110-ζ11 | orthogonal lifted from D22 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | ζ117+ζ114 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | ζ116+ζ115 | ζ118+ζ113 | -ζ1110-ζ11 | ζ1110+ζ11 | orthogonal lifted from D22 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | ζ119+ζ112 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | ζ118+ζ113 | ζ117+ζ114 | -ζ116-ζ115 | ζ116+ζ115 | orthogonal lifted from D22 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ119-ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ116-ζ115 | orthogonal lifted from D22 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ118+ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | ζ118+ζ113 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | ζ1110+ζ11 | ζ116+ζ115 | -ζ119-ζ112 | ζ119+ζ112 | orthogonal lifted from D22 |
| ρ23 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ116+ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ116-ζ115 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ119-ζ112 | ζ118+ζ113 | -ζ118-ζ113 | orthogonal lifted from D22 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ119-ζ112 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ118-ζ113 | ζ1110+ζ11 | -ζ1110-ζ11 | orthogonal lifted from D22 |
| ρ26 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ1110+ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ116-ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ117-ζ114 | orthogonal lifted from D22 |
| ρ28 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ117+ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | orthogonal lifted from D11 |
►On 44 points
Generators in S44
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(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)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)
G:=sub<Sym(44)| (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,33)(20,32)(21,31)(22,30)>;
G:=Group( (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,33)(20,32)(21,31)(22,30) );
G=PermutationGroup([[(1,43),(2,44),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,44),(11,43),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,33),(20,32),(21,31),(22,30)]])
C22×D11 is a maximal subgroup of
D22⋊C4
C22×D11 is a maximal quotient of D44⋊5C2 D4⋊2D11 D44⋊C2
Matrix representation of C22×D11 ►in GL3(𝔽23) generated by
| 1 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 22 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 22 | 4 |
| 22 | 0 | 0 |
| 0 | 0 | 22 |
| 0 | 22 | 0 |
G:=sub<GL(3,GF(23))| [1,0,0,0,22,0,0,0,22],[22,0,0,0,1,0,0,0,1],[1,0,0,0,0,22,0,1,4],[22,0,0,0,0,22,0,22,0] >;
C22×D11 in GAP, Magma, Sage, TeX
C_2^2\times D_{11} % in TeX
G:=Group("C2^2xD11"); // GroupNames label
G:=SmallGroup(88,11);
// by ID
G=gap.SmallGroup(88,11);
# by ID
G:=PCGroup([4,-2,-2,-2,-11,1283]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C22×D11 in TeX
Character table of C22×D11 in TeX