Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13466 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #2065 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
314 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)(3,16,11,23)(4,20,25,18)(5,21)(6,27)(7,12)(8,30)(9,15)(10,17,24,31)(13,26)(14,29)(19,28,32,22)(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);(1,3,5,17,8,19,7,18,9,11,26,31,29,32,6,20)(2,10,12,16,14,4,21,22,15,24,27,23,30,25,13,28);(1,4)(2,11)(3,15)(5,23)(6,28)(7,22)(8,10)(9,25)(12,20)(13,17)(14,19)(16,26)(18,27)(21,31)(24,29)(30,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6454);