Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C27 |
| direct product | C53×#23399 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,17)(15,19)(18,20)(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);(6,10)(8,16)(11,17)(12,13)(14,20)(18,19);(2,4)(8,16)(10,12)(14,20)(17,18);(5,7,9,15)(6,11,13,19)(8,14,16,20)(10,17,12,18)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16387);