Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#33470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#442 |
| commutator | C8 | C23×C4×C11 |
| Frattini, cosocle | C2×C8 | C24×C11 |
| prime radical | C2×C11×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ C11×#405 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,25,29,55)(15,31)(16,21)(17,38)(18,24)(19,28,44,63)(20,34)(22,36)(23,37)(26,56,59,75)(27,54,30,49)(32,46)(33,50)(35,47)(39,52,64,43)(40,45)(41,48)(42,68,73,71)(51,61)(53,60,72,57)(58,74,62,65)(66,69)(67,70);(12,14)(13,19)(15,26)(16,27)(17,39)(18,29)(20,42)(21,43)(22,49)(23,52)(24,44)(25,48)(28,61)(30,37)(31,65)(32,57)(33,68)(34,60)(35,59)(36,63)(38,55)(40,71)(41,64)(45,56)(46,73)(47,74)(50,75)(51,54)(53,66)(58,67)(62,69)(70,72);(12,15,16,34,18,35,37,32)(13,20,23,31,24,46,21,47)(14,26,27,60,29,59,30,57)(17,40,36,67,41,33,61,69)(19,42,52,65,44,73,43,74)(22,50,38,70,51,45,48,66)(25,53,49,75,55,72,54,56)(28,62,39,71,63,58,64,68);(12,16,18,37)(13,21,24,23)(14,27,29,30)(15,32,35,34)(17,36,41,61)(19,43,44,52)(20,31,46,47)(22,48,51,38)(25,54,55,49)(26,57,59,60)(28,39,63,64)(33,67,40,69)(42,65,73,74)(45,66,50,70)(53,75,72,56)(58,71,62,68)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 26458);