Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| Sylow | C53 ⋊ #14679 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D53 |
| commutator | C23×C53 | C23×C4 |
| Fitting, prime radical | C27×C53 | C2 |
| socle, cosocle | C24×C53 | C24 |
| hypercenter | C27 | D53 |
| hypocenter | C53 | #14679 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C27 | |
30862 conjugacy classes of subgroups; only the 1323 normal subgroups are shown.
This group has 30862 conjugacy classes of subgroups — too many to draw. Only its 1323 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) = 1696 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(67,69);(63,65);(59,61);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)Polycyclic presentation:
2^8,53|a^2=h,b^a=be,c^a=cf,d^a=dg,i^a=i^523416 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 100724);