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×#12756 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1790 |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,7,13,4,10)(3,14,15,8,17,24,21,25)(5,11,19,27,20,26,18,9)(16,29,30,23,32,22,31,28)(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)(2,8)(4,15)(5,16)(6,21)(7,17)(9,22)(10,14)(11,28)(12,24)(13,25)(18,30)(19,31)(20,32)(23,26)(27,29);(1,4,7,6)(2,9,13,27)(3,15,17,21)(5,18,20,19)(8,22,25,29)(10,26,12,11)(14,23,24,28)(16,30,32,31)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5744);