Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#12597 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | #117 |
| commutator | C2×C4 | C2×C42×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C2×C11 | #1746 |
| prime radical | C23×C11 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C11×#405 ◁ G | |
258 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)(12,13,16,21,17,23,33,45,18,22,34,44,35,46,60,70)(14,24,27,48,28,49,53,72,26,19,36,38,52,39,62,65)(15,20,30,41,32,42,58,68,31,43,57,69,59,63,74,75)(25,47,51,71,29,54,56,73,50,37,61,64,55,40,66,67);(12,14)(13,19,22,24)(15,29,31,55)(16,27)(17,28)(18,26)(20,40)(21,38,44,48)(23,39,46,49)(25,32,50,59)(30,56,57,66)(33,53)(34,36)(35,52)(37,63)(41,67)(42,47)(43,54)(45,65,70,72)(51,58,61,74)(60,62)(64,75)(68,71)(69,73);(12,15)(13,20)(14,25,26,50)(16,30)(17,32)(18,31)(19,37,24,47)(21,41)(22,43)(23,42)(27,51,36,61)(28,29,52,55)(33,58)(34,57)(35,59)(38,64,48,71)(39,40,49,54)(44,69)(45,68)(46,63)(53,56,62,66)(60,74)(65,67,72,73)(70,75)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 5585);