Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#13071 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #466 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C2×C11 | #1893 |
| prime radical | C11×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C2×C11×#131 ◁ G | |
456 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)(14,25,30,50)(15,31,34,49)(16,35)(17,23)(18,42)(19,24)(20,28,45,36)(21,33,48,41)(22,26)(27,58)(29,54,40,55)(32,47,37,51)(38,52)(39,68)(43,69)(44,64,46,56)(53,61)(57,73,67,66)(59,71)(60,70)(62,74)(63,65)(72,75);(12,14,26,56,17,40,16,36,19,30,38,64,18,29,39,28)(13,20,35,54,42,44,22,50,24,45,68,55,23,46,52,25)(15,27,57,69,41,60,37,59,34,62,67,75,33,61,32,63)(21,43,66,74,31,65,51,70,48,72,73,58,49,71,47,53);(12,15)(13,21)(14,27)(16,37)(17,41)(18,33)(19,34)(20,43)(22,51)(23,49)(24,48)(25,53)(26,57)(28,63)(29,61)(30,62)(31,42)(32,39)(35,66)(36,59)(38,67)(40,60)(44,65)(45,72)(46,71)(47,52)(50,70)(54,74)(55,58)(56,69)(64,75)(68,73)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6059);