Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#2949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D4 |
| commutator | C4 | C23×C4×C7 |
| Frattini, cosocle | C2×C4 | C24×C7 |
| prime radical | C22×C7×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C7×#117 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13)(14,20)(15,23)(16,18)(17,21)(19,25)(22,27)(24,26);(12,14,16,22)(13,17,18,26)(15,24,25,21)(19,27,23,20);(8,10,11,9)(12,13)(14,17)(15,19)(16,18)(20,21)(22,26)(23,25)(24,27);(13,18)(17,26)(19,23)(20,27)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17024);