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) = C22 |
| direct product | C7×#1482 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C42 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C4 |
| prime radical | C7×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23)(24,25,27,30,28,31,34,37)(26,29,33,36,32,35,38,39);(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20)(24,26)(25,29)(27,33)(28,32)(30,36)(31,35)(34,38)(37,39);(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17)(26,32)(29,35)(33,38)(36,39)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4717);