Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#8178 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C2×D4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle, prime radical | C23×C53 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C43×C53 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,4,3)(5,6)(7,9)(8,12)(10,11)(13,14)(15,18)(16,20)(17,19)(21,25)(22,24)(23,26)(27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,7,9,6,10,12,8,11)(13,15,16,21,17,22,23,27)(14,18,20,25,19,24,26,28);(1,3,4,2)(6,11)(7,12)(14,19)(18,24)(20,26)(25,28)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1166);