Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1804 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,7,13,4,10)(3,14,15,27,18,32,17,8)(5,20,23,11,24,31,19,25)(9,28,29,21,26,22,30,16)(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)(2,8,13,27)(4,15)(5,21)(6,17)(7,18)(9,25,26,11)(10,14,12,32)(16,24)(19,22)(20,29,31,30)(23,28);(1,3)(2,9)(4,17)(5,22)(6,15)(7,18)(8,25)(10,30)(11,27)(12,29)(13,26)(14,20)(16,23)(19,21)(24,28)(31,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5816);