Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#13069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #1893 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| prime radical | C2×C11×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ C2×C11×#131 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,25)(15,31,34,49)(16,35)(17,23)(18,41)(19,24)(20,28)(21,33,48,40)(22,37)(26,51)(27,58,61,74)(29,53)(30,55)(32,47,36,50)(38,42)(39,54)(43,70,73,75)(44,64)(45,56)(46,65)(52,60,71,62)(57,67,68,69)(59,66,63,72);(12,14,26,56,18,29,16,28,19,30,37,64,17,39,38,65)(13,20,42,53,23,45,22,25,24,46,35,54,41,44,51,55)(15,27,57,70,33,62,36,63,34,61,68,75,40,60,32,59)(21,43,69,74,49,72,50,52,48,73,67,58,31,66,47,71);(12,15)(13,21)(14,27)(16,36)(17,40)(18,33)(19,34)(20,43)(22,50)(23,49)(24,48)(25,52)(26,57)(28,63)(29,62)(30,61)(31,41)(32,38)(35,67)(37,68)(39,60)(42,69)(44,66)(45,72)(46,73)(47,51)(53,74)(54,58)(55,71)(56,70)(59,65)(64,75)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6057);