direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D11, C44⋊2C2, D22.C2, C4○Dic11, C2.1D22, Dic11⋊2C2, C22.2C22, C11⋊1(C2×C4), SmallGroup(88,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C4×D11 |
Generators and relations for C4×D11
G = < a,b,c | a4=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C4×D11
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | 44A | 44B | 44C | 44D | 44E | 44F | 44G | 44H | 44I | 44J | |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 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 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ1110+ζ11 | ζ118+ζ113 | ζ118+ζ113 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ116+ζ115 | ζ117+ζ114 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | orthogonal lifted from D22 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ117+ζ114 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | orthogonal lifted from D22 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ119+ζ112 | ζ116+ζ115 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | orthogonal lifted from D22 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ117+ζ114 | ζ1110+ζ11 | ζ1110+ζ11 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ118+ζ113 | ζ119+ζ112 | ζ119+ζ112 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | orthogonal lifted from D11 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ116+ζ115 | ζ117+ζ114 | ζ117+ζ114 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ16 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ1110+ζ11 | ζ118+ζ113 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | orthogonal lifted from D22 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ119+ζ112 | ζ116+ζ115 | ζ116+ζ115 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ18 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ118+ζ113 | ζ119+ζ112 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | orthogonal lifted from D22 |
| ρ19 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ119-ζ112 | ζ43ζ119+ζ43ζ112 | ζ4ζ119+ζ4ζ112 | ζ4ζ117+ζ4ζ114 | ζ4ζ116+ζ4ζ115 | ζ4ζ118+ζ4ζ113 | ζ4ζ1110+ζ4ζ11 | ζ43ζ117+ζ43ζ114 | ζ43ζ116+ζ43ζ115 | ζ43ζ118+ζ43ζ113 | ζ43ζ1110+ζ43ζ11 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ117-ζ114 | ζ43ζ117+ζ43ζ114 | ζ4ζ117+ζ4ζ114 | ζ4ζ118+ζ4ζ113 | ζ4ζ1110+ζ4ζ11 | ζ4ζ116+ζ4ζ115 | ζ4ζ119+ζ4ζ112 | ζ43ζ118+ζ43ζ113 | ζ43ζ1110+ζ43ζ11 | ζ43ζ116+ζ43ζ115 | ζ43ζ119+ζ43ζ112 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ116-ζ115 | ζ4ζ116+ζ4ζ115 | ζ43ζ116+ζ43ζ115 | ζ43ζ1110+ζ43ζ11 | ζ43ζ117+ζ43ζ114 | ζ43ζ119+ζ43ζ112 | ζ43ζ118+ζ43ζ113 | ζ4ζ1110+ζ4ζ11 | ζ4ζ117+ζ4ζ114 | ζ4ζ119+ζ4ζ112 | ζ4ζ118+ζ4ζ113 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ116-ζ115 | ζ43ζ116+ζ43ζ115 | ζ4ζ116+ζ4ζ115 | ζ4ζ1110+ζ4ζ11 | ζ4ζ117+ζ4ζ114 | ζ4ζ119+ζ4ζ112 | ζ4ζ118+ζ4ζ113 | ζ43ζ1110+ζ43ζ11 | ζ43ζ117+ζ43ζ114 | ζ43ζ119+ζ43ζ112 | ζ43ζ118+ζ43ζ113 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ117-ζ114 | ζ4ζ117+ζ4ζ114 | ζ43ζ117+ζ43ζ114 | ζ43ζ118+ζ43ζ113 | ζ43ζ1110+ζ43ζ11 | ζ43ζ116+ζ43ζ115 | ζ43ζ119+ζ43ζ112 | ζ4ζ118+ζ4ζ113 | ζ4ζ1110+ζ4ζ11 | ζ4ζ116+ζ4ζ115 | ζ4ζ119+ζ4ζ112 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ118-ζ113 | ζ43ζ118+ζ43ζ113 | ζ4ζ118+ζ4ζ113 | ζ4ζ116+ζ4ζ115 | ζ4ζ119+ζ4ζ112 | ζ4ζ1110+ζ4ζ11 | ζ4ζ117+ζ4ζ114 | ζ43ζ116+ζ43ζ115 | ζ43ζ119+ζ43ζ112 | ζ43ζ1110+ζ43ζ11 | ζ43ζ117+ζ43ζ114 | complex faithful |
| ρ25 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ118-ζ113 | ζ4ζ118+ζ4ζ113 | ζ43ζ118+ζ43ζ113 | ζ43ζ116+ζ43ζ115 | ζ43ζ119+ζ43ζ112 | ζ43ζ1110+ζ43ζ11 | ζ43ζ117+ζ43ζ114 | ζ4ζ116+ζ4ζ115 | ζ4ζ119+ζ4ζ112 | ζ4ζ1110+ζ4ζ11 | ζ4ζ117+ζ4ζ114 | complex faithful |
| ρ26 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ1110-ζ11 | ζ43ζ1110+ζ43ζ11 | ζ4ζ1110+ζ4ζ11 | ζ4ζ119+ζ4ζ112 | ζ4ζ118+ζ4ζ113 | ζ4ζ117+ζ4ζ114 | ζ4ζ116+ζ4ζ115 | ζ43ζ119+ζ43ζ112 | ζ43ζ118+ζ43ζ113 | ζ43ζ117+ζ43ζ114 | ζ43ζ116+ζ43ζ115 | complex faithful |
| ρ27 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ1110-ζ11 | ζ4ζ1110+ζ4ζ11 | ζ43ζ1110+ζ43ζ11 | ζ43ζ119+ζ43ζ112 | ζ43ζ118+ζ43ζ113 | ζ43ζ117+ζ43ζ114 | ζ43ζ116+ζ43ζ115 | ζ4ζ119+ζ4ζ112 | ζ4ζ118+ζ4ζ113 | ζ4ζ117+ζ4ζ114 | ζ4ζ116+ζ4ζ115 | complex faithful |
| ρ28 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ119-ζ112 | ζ4ζ119+ζ4ζ112 | ζ43ζ119+ζ43ζ112 | ζ43ζ117+ζ43ζ114 | ζ43ζ116+ζ43ζ115 | ζ43ζ118+ζ43ζ113 | ζ43ζ1110+ζ43ζ11 | ζ4ζ117+ζ4ζ114 | ζ4ζ116+ζ4ζ115 | ζ4ζ118+ζ4ζ113 | ζ4ζ1110+ζ4ζ11 | complex faithful |
►On 44 points
Generators in S44
(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)
G:=sub<Sym(44)| (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)>;
G:=Group( (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43) );
G=PermutationGroup([[(1,43,21,32),(2,44,22,33),(3,34,12,23),(4,35,13,24),(5,36,14,25),(6,37,15,26),(7,38,16,27),(8,39,17,28),(9,40,18,29),(10,41,19,30),(11,42,20,31)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43)]])
C4×D11 is a maximal subgroup of
C88⋊C2 D44⋊5C2 D4⋊2D11 D44⋊C2 D33⋊C4 D55⋊2C4
C4×D11 is a maximal quotient of C88⋊C2 Dic11⋊C4 D22⋊C4 D33⋊C4 D55⋊2C4
Matrix representation of C4×D11 ►in GL3(𝔽89) generated by
| 34 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 88 | 71 |
| 88 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(89))| [34,0,0,0,88,0,0,0,88],[1,0,0,0,0,88,0,1,71],[88,0,0,0,0,1,0,1,0] >;
C4×D11 in GAP, Magma, Sage, TeX
C_4\times D_{11} % in TeX
G:=Group("C4xD11"); // GroupNames label
G:=SmallGroup(88,4);
// by ID
G=gap.SmallGroup(88,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-11,21,1283]);
// Polycyclic
G:=Group<a,b,c|a^4=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×D11 in TeX
Character table of C4×D11 in TeX