Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#11992 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #1555 |
| commutator | C42 | C22×C4×C11 |
| Frattini, cosocle | C4×C8 | C23×C11 |
| prime radical | C11×#3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C11×M16 ◁ G | |
412 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,20,15,19,23,27,16,18,22,26,24,17,21,25);(13,17)(14,21)(15,24)(18,19)(20,26)(22,23);(13,18)(14,22)(17,19)(21,23)308 conjugacy classes — view the full character table.
| Fitting subgroup of | D11×#11992, #72269664 |
| maximal subgroup of | D11×#11992, C11×#489347, #72269664 |
| maximal normal subgroup of | D11×#11992, C11×#489347, #72269664 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 4980);