Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C53 · Schur multiplier (H2) = C213 |
| direct product | C2×C53×#3458 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C53 |
| socle | C23×C53 | C25 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
7022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 6 generators on 79 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(56,57,59,61)(58,62,63,60)(64,65)(66,72)(67,71)(68,70)(69,73)(74,78)(75,77)(76,79);(64,66)(65,69)(67,74)(68,73)(70,72)(71,76)(75,79)(77,78);(56,58,59,63)(57,60,61,62)(66,73)(69,72)(74,79)(76,78);(65,70)(69,72)(71,77)(76,78);(64,67,68,75)(65,71,70,77)(66,74,73,79)(69,76,72,78)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55672);