Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #120 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #386 |
| prime radical | C7×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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)(10,19,22,30)(11,23)(12,17)(13,27)(14,18)(15,21,31,26)(16,24)(20,35,38,29)(25,32)(28,33)(34,36,39,37);(8,10,12,26,14,22,13,21)(9,15,17,30,18,31,27,19)(11,20,24,37,25,38,28,36)(16,29,32,39,33,35,23,34);(8,11)(9,16)(10,20)(12,24)(13,28)(14,25)(15,29)(17,32)(18,33)(19,34)(21,36)(22,38)(23,27)(26,37)(30,39)(31,35)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5116);