Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#11211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4)(6,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,24)(22,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(5,6,8,7)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23);(5,7,8,6)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4199);