Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C8 · Schur multiplier (H2) = C215 |
| direct product | C24×D53⋊C8 |
| unambiguous semidirect product | D53⋊(C24×C8) |
| derived, Sylow | C53 ⋊ C25×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | #6157 |
| commutator, hypocenter | C53 | C25×C8 |
| Frattini | C2 | C25×#6157 |
| Fitting | C24×C4×C53 | C4 |
| socle | C25×C53 | C2×C4 |
| prime radical | C25×D53 | C4 |
| cosocle | Dic53 | C26 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C24×C4 | |
15454 conjugacy classes of subgroups; only the 8408 normal subgroups are shown.
This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 8408 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 6 generators on 69 points:
(1,2,5,10,19,32,21,8,16,27,39,48,51,44,30,42,28,40,49,45,35,22,13,6,11,20,33,41,25,12,3,7,14,23,36,46,34,17,29,43,50,53,52,47,38,26,15,24,37,31,18,9,4)(62,64,66,65)(63,67,69,68);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53)(62,63,65,68,66,69,64,67);(60,61);(58,59);(56,57);(54,55)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1116291);