Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #397 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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)(10,18,21,31)(11,15)(12,16)(13,17)(14,20,28,25)(19,34,22,33)(23,29)(24,30)(26,32)(27,36,38,35)(37,39);(8,10)(9,14)(11,22)(12,25)(13,21)(15,27)(16,31)(17,28)(18,32)(19,24)(20,26)(23,35)(29,34)(30,38)(33,39)(36,37);(8,11,12,23,13,24,26,37)(9,15,16,29,17,30,32,39)(10,19,20,35,21,22,25,36)(14,27,18,33,28,38,31,34)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5315);