Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#24336 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C4×Q8×C53 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(14,19)(16,17)(20,22)(21,26)(23,24)(27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,7,9,8)(6,10,12,11)(14,20)(16,17)(19,22)(21,27)(23,24)(26,28);(2,4)(13,14,18,22)(15,21,25,28)(16,20,17,19)(23,27,24,26);(5,8,9,7)(6,11,12,10)(13,15,18,25)(14,21,22,28)(16,23,17,24)(19,26,20,27)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 17324);