Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12757 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1790 |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10,7,11,4,17)(3,12,15,8,16,23,13,24)(5,9,20,26,21,27,18,25)(14,28,30,22,19,31,29,32)(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);(1,3,7,16)(2,8,11,24)(4,13,6,15)(5,19,21,14)(9,22,27,32)(10,12,17,23)(18,30,20,29)(25,31,26,28);(2,9)(8,22)(10,26)(11,27)(12,28)(17,25)(23,31)(24,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5745);