Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C2×C42 |
| direct product | C53×#23351 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C22×D4×C53 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,22,25,21,23,20,28)(19,24,26,32,27,30,29,31)(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);(3,8)(5,7)(9,10)(11,16)(12,13)(14,15)(18,23)(24,30)(25,28)(31,32);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,19)(18,24)(20,29)(21,27)(22,26)(23,30)(25,32)(28,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16339);