Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#20679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3,4)(5,6,10,13,11,8,9,19)(7,14,18,15,16,20,17,12)(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);(5,7,9,17,11,16,10,18)(6,12,13,14,8,15,19,20);(5,8)(6,11)(7,15)(9,19)(10,13)(12,16)(14,18)(17,20);(7,16)(12,15)(14,20)(17,18)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13667);