Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#3293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,15,20)(10,21,24,26)(11,23,16,30)(12,17,29,35)(13,19,14,18)(22,37,39,33)(25,38,32,27)(28,36,31,34);(8,10,15,24)(9,16,20,11)(12,27,29,38)(13,30,14,23)(17,33,35,37)(18,26,19,21)(22,31,39,28)(25,34,32,36);(8,11,15,16)(9,10,20,24)(12,25,29,32)(13,26,14,21)(17,22,35,39)(18,23,19,30)(27,36,38,34)(28,37,31,33);(8,12)(9,17)(10,22)(11,25)(13,31)(14,28)(15,29)(16,32)(18,36)(19,34)(20,35)(21,37)(23,38)(24,39)(26,33)(27,30)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17368);