Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12755 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1790 |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3537 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,5,7,6,9,3,10)(4,8,13,14,12,16,11,15)(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);(2,7)(4,12)(8,14)(9,10)(11,13)(15,16);(2,8)(7,14)(9,16)(10,15)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5743);