Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#12366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C23×C11 |
| prime radical | C11×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
364 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 78 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)(13,17,19,27)(14,21)(15,24)(18,23,20,26)(22,25);(14,22)(15,24)(17,23)(18,20)(21,25)(26,27);(12,13)(14,23)(15,18)(16,19)(17,22)(20,24)(21,27)(25,26)308 conjugacy classes — view the full character table.
| Fitting subgroup of | D11×#12366, #72269632 |
| maximal subgroup of | D11×#12366, C11×#489347, #72269632 |
| maximal normal subgroup of | D11×#12366, C11×#489347, #72269632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 5354);