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×#61096 |
| 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×#532 ◁ G | |
2686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 89 points:
(1,2)(3,4)(5,6,10,15)(7,12,18,26)(8,19,20,13)(9,11,22,14)(16,29,23,24)(17,31,33,27)(21,28,34,25)(30,35,32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(6,11)(7,16)(8,20)(9,22)(12,26)(13,28)(14,15)(17,32)(18,23)(19,25)(30,33)(35,36);(2,4)(8,20)(13,19)(17,33)(21,34)(25,28)(27,31)(30,32)(35,36);(5,7,10,18)(6,12,15,26)(8,17,20,33)(9,23,22,16)(11,24,14,29)(13,27,19,31)(21,32,34,30)(25,35,28,36);(5,8)(6,13)(7,17)(9,21)(10,20)(11,25)(12,27)(14,28)(15,19)(16,30)(18,33)(22,34)(23,32)(24,35)(26,31)(29,36)2915 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54084);