Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19792 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,9)(3,6,11,15)(4,7,12,8)(10,13,16,14)(18,24)(19,26)(21,22)(25,32)(27,28)(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);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,18,23,19)(20,28,30,32)(21,26,22,24)(25,31,27,29);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,29,32,31);(17,20)(18,25)(19,27)(21,31)(22,29)(23,30)(24,32)(26,28)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12780);