Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #386 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 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)(8,9,12,17,13,16,18,31)(10,14,21,28,20,29,25,32)(11,22,24,33,23,15,26,30)(19,27,35,38,34,37,36,39);(8,10,18,21,13,20,12,25)(9,14,17,32,16,29,31,28)(11,19,24,36,23,34,26,35)(15,27,33,38,22,37,30,39);(8,11)(9,15)(10,19)(12,26)(13,23)(14,27)(16,22)(17,33)(18,24)(20,34)(21,36)(25,35)(28,39)(29,37)(30,31)(32,38)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5150);