metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C11⋊2D4, C22⋊D11, D22⋊2C2, Dic11⋊C2, C2.5D22, C22.5C22, (C2×C22)⋊2C2, SmallGroup(88,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11⋊D4
G = < a,b,c | a11=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C11⋊D4
| class | 1 | 2A | 2B | 2C | 4 | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | 22F | 22G | 22H | 22I | 22J | 22K | 22L | 22M | 22N | 22O | |
| size | 1 | 1 | 2 | 22 | 22 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ117-ζ114 | ζ118+ζ113 | ζ1110+ζ11 | orthogonal lifted from D22 |
| ρ7 | 2 | 2 | 2 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | orthogonal lifted from D11 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ118-ζ113 | ζ116+ζ115 | ζ119+ζ112 | orthogonal lifted from D22 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ116-ζ115 | ζ1110+ζ11 | ζ117+ζ114 | orthogonal lifted from D22 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ119-ζ112 | ζ117+ζ114 | ζ116+ζ115 | orthogonal lifted from D22 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ1110-ζ11 | ζ119+ζ112 | ζ118+ζ113 | orthogonal lifted from D22 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ117+ζ114 | ζ118-ζ113 | -ζ1110+ζ11 | -ζ116+ζ115 | ζ119-ζ112 | -ζ119+ζ112 | ζ116-ζ115 | ζ1110-ζ11 | -ζ118+ζ113 | ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | complex faithful |
| ρ17 | 2 | -2 | 0 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ119-ζ112 | ζ119-ζ112 | ζ117-ζ114 | -ζ116+ζ115 | -ζ118+ζ113 | -ζ1110+ζ11 | ζ1110-ζ11 | ζ118-ζ113 | ζ116-ζ115 | -ζ117+ζ114 | -ζ119+ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ117-ζ114 | ζ117-ζ114 | -ζ118+ζ113 | ζ1110-ζ11 | ζ116-ζ115 | -ζ119+ζ112 | ζ119-ζ112 | -ζ116+ζ115 | -ζ1110+ζ11 | ζ118-ζ113 | -ζ117+ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ1110+ζ11 | -ζ119+ζ112 | -ζ118+ζ113 | -ζ117+ζ114 | -ζ116+ζ115 | ζ116-ζ115 | ζ117-ζ114 | ζ118-ζ113 | ζ119-ζ112 | ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ1110-ζ11 | ζ1110-ζ11 | ζ119-ζ112 | ζ118-ζ113 | ζ117-ζ114 | ζ116-ζ115 | -ζ116+ζ115 | -ζ117+ζ114 | -ζ118+ζ113 | -ζ119+ζ112 | -ζ1110+ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ116-ζ115 | ζ116-ζ115 | -ζ1110+ζ11 | ζ117-ζ114 | -ζ119+ζ112 | ζ118-ζ113 | -ζ118+ζ113 | ζ119-ζ112 | -ζ117+ζ114 | ζ1110-ζ11 | -ζ116+ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ119+ζ112 | -ζ117+ζ114 | ζ116-ζ115 | ζ118-ζ113 | ζ1110-ζ11 | -ζ1110+ζ11 | -ζ118+ζ113 | -ζ116+ζ115 | ζ117-ζ114 | ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ118+ζ113 | ζ116-ζ115 | ζ119-ζ112 | -ζ1110+ζ11 | -ζ117+ζ114 | ζ117-ζ114 | ζ1110-ζ11 | -ζ119+ζ112 | -ζ116+ζ115 | ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ116+ζ115 | ζ1110-ζ11 | -ζ117+ζ114 | ζ119-ζ112 | -ζ118+ζ113 | ζ118-ζ113 | -ζ119+ζ112 | ζ117-ζ114 | -ζ1110+ζ11 | ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | complex faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ118-ζ113 | ζ118-ζ113 | -ζ116+ζ115 | -ζ119+ζ112 | ζ1110-ζ11 | ζ117-ζ114 | -ζ117+ζ114 | -ζ1110+ζ11 | ζ119-ζ112 | ζ116-ζ115 | -ζ118+ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | complex faithful |
►On 44 points
Generators in S44
(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 32 21 43)(2 31 22 42)(3 30 12 41)(4 29 13 40)(5 28 14 39)(6 27 15 38)(7 26 16 37)(8 25 17 36)(9 24 18 35)(10 23 19 34)(11 33 20 44)
(2 11)(3 10)(4 9)(5 8)(6 7)(12 19)(13 18)(14 17)(15 16)(20 22)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 44)(32 43)(33 42)
G:=sub<Sym(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), (1,32,21,43)(2,31,22,42)(3,30,12,41)(4,29,13,40)(5,28,14,39)(6,27,15,38)(7,26,16,37)(8,25,17,36)(9,24,18,35)(10,23,19,34)(11,33,20,44), (2,11)(3,10)(4,9)(5,8)(6,7)(12,19)(13,18)(14,17)(15,16)(20,22)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,44)(32,43)(33,42)>;
G:=Group( (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), (1,32,21,43)(2,31,22,42)(3,30,12,41)(4,29,13,40)(5,28,14,39)(6,27,15,38)(7,26,16,37)(8,25,17,36)(9,24,18,35)(10,23,19,34)(11,33,20,44), (2,11)(3,10)(4,9)(5,8)(6,7)(12,19)(13,18)(14,17)(15,16)(20,22)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,44)(32,43)(33,42) );
G=PermutationGroup([[(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)], [(1,32,21,43),(2,31,22,42),(3,30,12,41),(4,29,13,40),(5,28,14,39),(6,27,15,38),(7,26,16,37),(8,25,17,36),(9,24,18,35),(10,23,19,34),(11,33,20,44)], [(2,11),(3,10),(4,9),(5,8),(6,7),(12,19),(13,18),(14,17),(15,16),(20,22),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,44),(32,43),(33,42)]])
C11⋊D4 is a maximal subgroup of
D44⋊5C2 D4×D11 D4⋊2D11 C33⋊D4 C11⋊D12 C33⋊7D4 C11⋊S4 C22⋊F11 C55⋊D4 C11⋊D20 C55⋊7D4
C11⋊D4 is a maximal quotient of
Dic11⋊C4 D22⋊C4 D4⋊D11 D4.D11 Q8⋊D11 C11⋊Q16 C23.D11 C33⋊D4 C11⋊D12 C33⋊7D4 C55⋊D4 C11⋊D20 C55⋊7D4
Matrix representation of C11⋊D4 ►in GL2(𝔽23) generated by
| 16 | 17 |
| 17 | 21 |
| 0 | 1 |
| 22 | 0 |
| 15 | 11 |
| 11 | 8 |
G:=sub<GL(2,GF(23))| [16,17,17,21],[0,22,1,0],[15,11,11,8] >;
C11⋊D4 in GAP, Magma, Sage, TeX
C_{11}\rtimes D_4 % in TeX
G:=Group("C11:D4"); // GroupNames label
G:=SmallGroup(88,7);
// by ID
G=gap.SmallGroup(88,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-11,49,1283]);
// Polycyclic
G:=Group<a,b,c|a^11=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C11⋊D4 in TeX
Character table of C11⋊D4 in TeX