Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C27 |
| direct product | C2×C53×#2018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G | |
2158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(3,4)(5,6)(7,8)(9,17)(10,15)(11,14)(12,16)(13,18)(19,22)(20,21);(4,6)(7,9)(8,13)(10,19)(11,18)(12,20)(14,21)(15,17)(16,22);(8,14)(9,18)(10,11)(15,16)(17,21)(19,20)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16423);