Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 101 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,22,25)(19,27,29,41)(20,24,32,39)(21,33,36,26)(23,34,38,30)(28,44,45,43)(31,42,35,40)(37,47,48,46)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(4,13)(8,12)(9,16)(11,14)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,38)(26,41)(27,33)(30,36)(31,47)(32,45)(35,46)(39,48)(40,43)(42,44);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20,22,32)(18,24,25,39)(19,28,29,45)(21,35,36,31)(23,37,38,48)(26,42,33,40)(27,43,41,44)(30,47,34,46);(2,7)(6,15)(8,12)(9,16)(18,25)(19,29)(24,39)(26,33)(28,45)(30,34)(40,42)(46,47)1643 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12077);