Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C7×#486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| prime radical | C2×C7×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
528 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4)(5,8)(6,7)(9,10)(12,17)(13,18)(14,15);(4,7)(8,10)(11,12,15,13,16,18,14,17);(3,5,6,9)(4,8,7,10)(11,13,14,12,16,17,15,18)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17203);