direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×D11, D55⋊C4, D5.1D22, C55⋊(C2×C4), C5⋊(C4×D11), C11⋊F5⋊C2, (C5×D11)⋊C4, (C11×F5)⋊C2, C11⋊1(C2×F5), (D5×D11).C2, (D5×C11).C22, SmallGroup(440,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — F5×D11 |
Generators and relations for F5×D11
G = < a,b,c,d | a5=b4=c11=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of F5×D11
►On 55 points
Generators in S55
(1 21 32 43 54)(2 22 33 44 55)(3 12 23 34 45)(4 13 24 35 46)(5 14 25 36 47)(6 15 26 37 48)(7 16 27 38 49)(8 17 28 39 50)(9 18 29 40 51)(10 19 30 41 52)(11 20 31 42 53)
(12 23 45 34)(13 24 46 35)(14 25 47 36)(15 26 48 37)(16 27 49 38)(17 28 50 39)(18 29 51 40)(19 30 52 41)(20 31 53 42)(21 32 54 43)(22 33 55 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)(45 46 47 48 49 50 51 52 53 54 55)
(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)(45 51)(46 50)(47 49)(52 55)(53 54)
G:=sub<Sym(55)| (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (12,23,45,34)(13,24,46,35)(14,25,47,36)(15,26,48,37)(16,27,49,38)(17,28,50,39)(18,29,51,40)(19,30,52,41)(20,31,53,42)(21,32,54,43)(22,33,55,44), (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)(45,46,47,48,49,50,51,52,53,54,55), (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)(45,51)(46,50)(47,49)(52,55)(53,54)>;
G:=Group( (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (12,23,45,34)(13,24,46,35)(14,25,47,36)(15,26,48,37)(16,27,49,38)(17,28,50,39)(18,29,51,40)(19,30,52,41)(20,31,53,42)(21,32,54,43)(22,33,55,44), (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)(45,46,47,48,49,50,51,52,53,54,55), (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)(45,51)(46,50)(47,49)(52,55)(53,54) );
G=PermutationGroup([[(1,21,32,43,54),(2,22,33,44,55),(3,12,23,34,45),(4,13,24,35,46),(5,14,25,36,47),(6,15,26,37,48),(7,16,27,38,49),(8,17,28,39,50),(9,18,29,40,51),(10,19,30,41,52),(11,20,31,42,53)], [(12,23,45,34),(13,24,46,35),(14,25,47,36),(15,26,48,37),(16,27,49,38),(17,28,50,39),(18,29,51,40),(19,30,52,41),(20,31,53,42),(21,32,54,43),(22,33,55,44)], [(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),(45,46,47,48,49,50,51,52,53,54,55)], [(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),(45,51),(46,50),(47,49),(52,55),(53,54)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 10 | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55E |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 |
| size | 1 | 5 | 11 | 55 | 5 | 5 | 55 | 55 | 4 | 44 | 2 | ··· | 2 | 10 | ··· | 10 | 10 | ··· | 10 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D11 | D22 | C4×D11 | F5 | C2×F5 | F5×D11 |
| kernel | F5×D11 | C11×F5 | C11⋊F5 | D5×D11 | C5×D11 | D55 | F5 | D5 | C5 | D11 | C11 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 1 | 1 | 5 |
Matrix representation of F5×D11 ►in GL6(𝔽661)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 245 |
| 0 | 0 | 2 | 1 | 1 | 245 |
| 0 | 0 | 1 | 2 | 1 | 245 |
| 0 | 0 | 634 | 634 | 634 | 657 |
| 106 | 0 | 0 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 660 | 660 | 660 | 416 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 218 | 219 | 0 | 0 | 0 | 0 |
| 660 | 660 | 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 | 219 | 0 | 0 | 0 | 0 |
| 0 | 660 | 0 | 0 | 0 | 0 |
| 0 | 0 | 660 | 0 | 0 | 0 |
| 0 | 0 | 0 | 660 | 0 | 0 |
| 0 | 0 | 0 | 0 | 660 | 0 |
| 0 | 0 | 0 | 0 | 0 | 660 |
G:=sub<GL(6,GF(661))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,1,634,0,0,1,1,2,634,0,0,1,1,1,634,0,0,245,245,245,657],[106,0,0,0,0,0,0,106,0,0,0,0,0,0,0,1,660,0,0,0,0,0,660,0,0,0,1,0,660,0,0,0,0,0,416,1],[218,660,0,0,0,0,219,660,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,219,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660] >;
F5×D11 in GAP, Magma, Sage, TeX
F_5\times D_{11} % in TeX
G:=Group("F5xD11"); // GroupNames label
G:=SmallGroup(440,43);
// by ID
G=gap.SmallGroup(440,43);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,26,168,173,10004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^11=d^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export