Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#24287 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,13)(4,18)(5,23)(6,25)(7,12)(8,15)(9,21)(10,19)(11,26)(14,31)(16,29)(17,28)(20,30)(22,27)(24,32)(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);(1,3,6,15,7,17,26,16)(2,8,11,13,12,29,25,28)(4,19,20,24,22,14,21,5)(9,10,27,23,30,31,18,32);(1,4)(2,9)(3,14)(5,15)(6,20)(7,22)(8,10)(11,27)(12,30)(13,23)(16,24)(17,19)(18,25)(21,26)(28,32)(29,31);(1,5,6,19,7,24,26,14)(2,10,11,23,12,31,25,32)(3,4,15,20,17,22,16,21)(8,27,13,30,29,18,28,9)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17275);