Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#575 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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)(8,9,13,17)(10,18,20,36)(11,22,24,38)(12,27,25,16)(14,29,30,19)(15,32,34,26)(21,37,28,33)(23,35,39,31);(8,10)(9,14)(11,23)(12,28)(13,20)(15,33)(16,35)(17,30)(18,22)(19,26)(21,25)(24,39)(27,31)(29,32)(34,37)(36,38);(11,24)(15,34)(18,36)(19,29)(22,38)(23,39)(26,32)(33,37)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5426);