Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8785 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle, prime radical | C23×C53 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C43×C53 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,3,5,4,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,23)(21,31)(22,26)(24,29)(25,28)(30,32)(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);(9,11,12,15)(10,13,14,16)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27);(10,14)(13,16)(17,20,21,29,22,30,28,19)(18,24,25,23,26,27,31,32)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1773);