Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12951 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1864 |
| 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 | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,13,8,14,7,24)(3,15,18,27,20,29,19,25)(4,12,22,30,23,31,5,11)(9,21,10,16,26,17,28,32)(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)(2,9,14,26)(4,21)(5,16)(6,19)(7,18)(8,20)(10,13,28,24)(11,27,30,25)(12,15,31,29)(17,23)(22,32);(1,3)(2,10)(4,21)(5,16)(6,19)(7,18)(8,20)(9,13)(11,29)(12,27)(14,28)(15,30)(17,23)(22,32)(24,26)(25,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5939);