Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1802 |
| 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 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,13,8,14,7,11)(3,12,6,9,17,25,16,26)(4,18,21,32,22,24,19,30)(10,27,28,20,29,23,31,15)(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);(1,3,5,6,8,17,7,16)(2,9,11,12,14,26,13,25)(4,15,19,23,22,20,21,27)(10,24,28,30,29,18,31,32);(1,4)(2,10)(3,15)(5,19)(6,23)(7,21)(8,22)(9,24)(11,28)(12,30)(13,31)(14,29)(16,27)(17,20)(18,26)(25,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5813);