Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#20639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×#117×C53 ◁ G | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(2,4)(5,6,9,14,10,13,22,27)(7,11,17,25,16,24,32,35)(8,18,19,28,20,12,21,26)(15,29,31,36,30,23,33,34)(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);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,23)(13,24)(14,25)(18,29)(19,31)(20,30)(21,33)(22,32)(26,34)(27,35)(28,36);(5,8,10,20)(6,12,13,18)(7,15,16,30)(9,21,22,19)(11,23,24,29)(14,28,27,26)(17,33,32,31)(25,36,35,34);(7,16)(11,24)(15,30)(17,32)(23,29)(25,35)(31,33)(34,36)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13627);