Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C24 |
| direct product | C2×C53×#1377 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C23×C53 | C22⋊C8 |
| prime radical | C24×C53 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C23×C8×C53 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 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);(60,61,62,66,63,67,69,73,64,65,68,71,70,72,74,75);(56,57,58,59)(61,65)(62,68)(67,72)(69,74)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5538);