Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#61077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C53×#525 ◁ G | |
2762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 73 points:
(1,2)(3,4)(5,6)(7,17)(8,14)(9,18)(10,12)(11,16)(13,19)(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);(6,12)(8,16)(9,20)(10,14)(13,15)(17,18);(2,4)(5,7,11,19)(6,13,16,17)(8,18,12,15)(9,14,20,10);(5,8,11,12)(6,14,16,10)(7,18,19,15)(9,13,20,17);(5,9)(6,15)(7,14)(8,13)(10,19)(11,20)(12,17)(16,18)2915 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54065);