Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| Fitting | C26×C53 ⋊ C4 |
| Sylow | C53 ⋊ #14679 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×#6157 |
| commutator, socle | C23×C53 | C23×C4 |
| Fitting | C26×C53 | C4 |
| prime radical | C26×D53 | C2 |
| cosocle | C23×D53 | C24 |
| hypercenter | C26 | #6157 |
| hypocenter | C53 | #14679 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C26 | |
30862 conjugacy classes of subgroups; only the 816 normal subgroups are shown.
This group has 30862 conjugacy classes of subgroups — too many to draw. Only its 816 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 65 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)(63,65);(59,61);(55,57);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)Polycyclic presentation:
2^8,53|a^2=h,b^a=be,c^a=cf,d^a=dg,i^a=i^30,i^h=i^52920 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1099174);