Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#7190 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C2×C4 | C4×C8×C53 |
| Frattini, cosocle | C2×#117 | C22×C53 |
| socle | C22×C53 | #342 |
| prime radical | C24×C53 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C2×#117×C53 ◁ G | |
322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 93 points:
(1,2,3,5,4,6,7,8)(9,10,13,18)(11,21,23,37)(12,19,26,35)(14,17,28,34)(15,20,29,36)(16,30,31,22)(24,33,38,40)(25,32,27,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(9,11,14,24,15,25,12,22)(10,16,19,32,20,33,17,21)(13,27,28,30,29,23,26,38)(18,31,35,39,36,40,34,37)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 178);