Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C2×#476 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,12)(4,18,22,24)(5,11)(6,13)(7,9)(8,14)(10,21,28,15)(16,26)(17,25)(19,32,20,29)(23,30,31,27)(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);(1,3,6,7,8,17,5,16)(2,9,11,12,14,26,13,25)(4,15,20,23,22,21,19,31)(10,24,27,29,28,18,30,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,23)(8,22)(9,24)(11,27)(12,29)(13,30)(14,28)(16,31)(17,21)(18,26)(25,32);(1,5,8,6)(2,11,14,13)(3,16,17,7)(4,19,22,20)(9,12,26,25)(10,27,28,30)(15,31,21,23)(18,32,24,29)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26625);