Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#33557 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C53 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G | |
1258 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)(3,7)(4,6)(5,8)(9,15)(10,12)(11,13)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(2,6)(3,9)(7,14)(8,13)(10,16)(12,15);(3,9)(4,11)(7,15)(12,14);(1,3)(2,7)(4,10)(5,9)(6,12)(8,15)(11,16)(13,14)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26545);