Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1878 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | C2×D4 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,15)(16,17,20,22,21,25,19,24)(18,26,28,23,29,30,27,31);(8,9,10,11)(17,22)(19,20)(23,26)(24,25)(27,28)(30,31);(13,15)(16,18,21,29)(17,23,25,31)(19,28,20,27)(22,26,24,30)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5113);