Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#10748 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #385 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6,9,14,10,13,22,28)(7,15,18,25,17,11,19,24)(8,12,21,26,20,27,33,36)(16,23,31,34,30,29,32,35)(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);(5,7,9,19,10,17,22,18)(6,11,14,25,13,15,28,24)(8,16,21,32,20,30,33,31)(12,23,26,35,27,29,36,34);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,27)(14,26)(15,29)(17,30)(18,31)(19,32)(22,33)(24,34)(25,35)(28,36)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3736);