Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#9716 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C2×#121×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 73 points:
(2,4)(5,6)(7,9,11,12)(8,10)(13,14,17,18,16,19,15,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,7)(6,9,10,12)(8,11)(14,18)(15,17)(19,20);(1,2,3,4)(6,10)(7,11)(14,19)(18,20)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2704);