Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9236 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #121 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #386 |
| prime radical | C23×C53 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,21,23,22,26,20,25)(19,27,29,24,30,31,28,32)(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)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(3,11)(7,15)(10,16)(12,13)(17,19,22,30)(18,24,26,32)(20,29,21,28)(23,27,25,31)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2224);