Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11307 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #397 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1443 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
548 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)(3,7,12,21)(4,14)(5,18)(6,9)(8,16)(10,19)(11,20,25,32)(13,27,26,22)(15,28,30,23)(17,24)(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);(1,3)(2,7,9,21)(4,15,16,30)(5,13)(6,12)(8,23)(10,22,18,27)(11,17,25,31)(14,28)(19,26)(20,29)(24,32);(1,4)(2,8,9,14)(3,11,12,25)(5,17)(6,16)(7,20)(10,24,18,29)(13,15,26,30)(19,31)(21,32)(22,23)(27,28)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4295);