Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C53×#1458 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C22⋊C4 |
| commutator | C22 | C23×C8×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C2×C22⋊C4 |
| prime radical | C25×C53 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C22×C42×C53 ◁ G | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 79 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);(56,57,58,60,59,61,62,63)(64,65)(66,72,74,69)(67,70)(68,71)(73,78,79,76)(75,77);(64,66)(65,69)(67,73)(68,74)(70,76)(71,72)(75,79)(77,78);(64,67,68,75)(65,70,71,77)(66,73,74,79)(69,76,72,78)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10333);