Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#3349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| prime radical | C2×C7×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×D4 ◁ G | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,12,11)(10,14,15,13)(16,17)(18,22)(19,26)(20,29)(21,24)(23,25)(27,31)(28,30);(8,10)(9,13)(11,14)(12,15)(16,18,21,20)(17,22,24,29)(19,27,28,25)(23,26,31,30);(8,11,12,9)(10,14,15,13)(16,19)(17,23)(18,25)(20,27)(21,28)(22,30)(24,31)(26,29);(10,15)(13,14)(16,18,21,20)(17,22,24,29)(19,25,28,27)(23,30,31,26)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17424);