Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C7 |
| prime radical | C7×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C7×D4 ◁ G | |
226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,15)(12,16)(13,17)(14,20)(19,35)(21,34)(22,33)(23,29)(24,30)(25,32)(26,36)(27,37)(28,38)(31,39);(8,10,13,21)(9,14,17,27)(11,22,24,19)(12,20,25,37)(15,28,30,26)(16,18,32,34)(23,38,39,36)(29,33,31,35);(8,11,13,24)(9,15,17,30)(10,19,21,22)(12,23,25,39)(14,26,27,28)(16,31,32,29)(18,33,34,35)(20,36,37,38)119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5436);