Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#11475 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | #99 |
| commutator | C2×C4 | C22×C8×C11 |
| Frattini, cosocle | C2×C4⋊C4 | C23×C11 |
| socle | C22×C11 | C2×#99 |
| prime radical | C2×C11×#127 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C4×C11 ◁ C2×C4×D4×C11 ◁ G | |
616 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,18,16,19,14,17)(20,21)(22,25,27,24)(23,26);(21,24)(22,27)(25,26);(12,14,16,15)(13,17,19,18)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 4463);