Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12754 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1790 |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C22×D4 | C23×C53 |
| prime radical | C53×#3533 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,29)(12,26)(14,25)(15,23,30,21)(16,22)(19,27,31,32)(24,28)(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)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,27)(16,28)(17,25)(18,29)(21,26)(22,31)(23,24)(30,32);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,15,18,30)(7,19,20,31)(9,23,17,21)(11,24,25,26)(13,27,29,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5742);