Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12918 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #120 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1850 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,11,5,12,22,30,7,13,24,31,23,10,28,29)(3,14,16,19,9,26,27,4,17,20,32,18,15,8,25,21)(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);(1,3,6,16,5,9,22,27,7,17,24,32,23,15,28,25)(2,8,11,21,12,14,30,19,13,26,31,4,10,20,29,18);(1,4)(2,9)(3,10)(5,18)(6,20)(7,21)(8,22)(11,27)(12,17)(13,15)(14,24)(16,29)(19,23)(25,31)(26,28)(30,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5906);