Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C234 |
| direct product | C23×#6342371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C25 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4473736019 conjugacy classes of subgroups; only the 1783035 normal subgroups are shown.
This group has 4473736019 conjugacy classes of subgroups — too many to draw. Only its 1783035 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 22 points:
(3,4);(8,10)(12,14)(16,18)(20,22);(20,22);(16,18);(5,6);(1,2)(3,4)(5,6);(12,14);(7,8,9,10)(11,12,13,14)(15,16,17,18)(19,20,21,22)1216 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#72232847, C2×#72296209, C23×#1202 |
| maximal normal subgroup of | C22×#72232847, C2×#72296209, C23×#1202 |
| maximal quotient of | C22×#72232847 |
| automorphism group of | #60313, #60316, #60322, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4), (8,10)(12,14)(16,18)(20,22), (20,22), (16,18), (5,6), (1,2)(3,4)(5,6), (12,14), (7,8,9,10)(11,12,13,14)(15,16,17,18)(19,20,21,22));