Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#1793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #120 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
552 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,17)(15,22)(18,26)(19,23)(20,24)(25,27);(12,13)(14,18)(15,23)(16,19)(17,22)(20,25)(21,27)(24,26);(12,14,16,21)(13,18,19,27)(15,24,22,20)(17,25,23,26)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5028);