Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#61874 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| prime radical | C53×#3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C53×#533 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,22)(19,23)(20,27)(25,31)(26,29)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,21,22)(19,23,26,29)(20,28,27,24)(25,31,32,30);(17,19,21,26)(18,23,22,29)(20,25,27,32)(24,30,28,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,18,21,22)(19,23,26,29)(20,24,27,28)(25,30,32,31);(2,8)(6,9)(7,16)(14,15)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,32)(29,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54862);