Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C8 |
| direct product | C53×#19656 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32)(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);(2,6)(7,14)(8,13)(9,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,30,31,32);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(19,25)(23,30)(26,31)(28,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12644);