Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13122 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C16×C53 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(2,3)(5,6,10,15,9,16,26,34,11,17,28,18,27,14,32,33)(7,12,20,29,19,23,13,30,21,8,22,24,35,25,31,36)(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);(1,2,4,3)(5,7,10,20,9,19,26,13,11,21,28,22,27,35,32,31)(6,12,15,29,16,23,34,30,17,8,18,24,14,25,33,36);(1,3,4,2)(5,8)(6,13)(7,18)(9,25)(10,24)(11,12)(14,20)(15,21)(16,22)(17,31)(19,33)(23,27)(26,36)(28,29)(30,32)(34,35)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6110);