Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C29 |
| direct product | C2×C53×#2906 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
2810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(57,59)(60,61,63,66,64,67,70,73)(62,65,69,72,68,71,74,75);(56,57,58,59);(60,62)(61,65)(63,69)(64,68)(66,72)(67,71)(70,74)(73,75);(62,68)(65,71)(69,74)(72,75)5300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53086);