Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C53 · Schur multiplier (H2) = C213 |
| direct product | C53×#62755 |
| 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 | |
6644 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,31)(30,32)(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)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31);(3,10)(6,9)(11,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(18,25)(23,26)(24,29)(31,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55743);