Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#23361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2380 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C53×#3151 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2)(3,10,12,7)(4,13,11,16)(5,8)(6,9)(14,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(1,3,6,12)(2,7,9,10)(4,14,11,5)(8,16,15,13);(1,4,5,3,6,11,14,12)(2,7,15,16,9,10,8,13);(2,8,9,15)(7,13,10,16)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16349);