Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#7258 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #102 |
| commutator | C2×C8 | C42×C53 |
| Frattini, cosocle | C23×C8 | C22×C53 |
| prime radical | C24×C53 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28)(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)(6,9,10,11)(8,12)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 246);