Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C53 · Schur multiplier (H2) = C213 |
| direct product | C53×#62752 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C53 |
| socle | C22×C53 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
6528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 6 generators on 85 points:
(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,16)(12,15)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,10)(2,6,7,13)(4,11,12,9)(8,16,15,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(2,7)(6,13)(8,15)(14,16)(19,26)(22,25)(27,32)(29,31);(1,4,5,12)(2,8,7,15)(3,9,10,11)(6,14,13,16)(18,23)(22,25)(24,30)(29,31);(2,7)(6,13)(8,15)(14,16);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55740);