Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#1390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C22×D4 | C23×C53 |
| prime radical | C23×D4×C53 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 67 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)(62,66,67,64)(63,65);(56,57,58,59)(61,64)(65,66)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5560);