Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C27 |
| direct product | C2×C53×#1878 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C2×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 79 points:
(1,2)(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);(3,4,6,5)(7,8,9,10)(11,12,15,17,16,20,14,19)(13,21,23,18,24,25,22,26);(3,5,6,4)(8,10)(12,17)(14,15)(18,21)(19,20)(22,23)(25,26);(11,13,16,24)(12,18,20,26)(14,23,15,22)(17,21,19,25)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13910);