Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12962 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | #1867 |
| prime radical | C53×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,11,7,12,18,27)(3,13,16,22,17,29,31,32)(4,10,5,9,19,25,21,26)(8,20,23,30,24,15,28,14)(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);(1,3,6,16,7,17,18,31)(2,8,11,23,12,24,27,28)(4,14,5,20,19,30,21,15)(9,22,25,29,26,32,10,13);(2,9)(8,22)(10,27)(11,25)(12,26)(13,28)(23,29)(24,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5950);