Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9007 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #386 |
| prime radical | C53×#3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 6784 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,5,7,4,8,3,6)(9,10,14,19,15,20,13,18)(11,16,22,30,23,31,28,34)(12,24,27,17,25,33,26,32)(21,35,37,40,36,39,38,29)(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);(1,2,4,8)(3,7,5,6)(9,11,14,22,15,23,13,28)(10,16,19,30,20,31,18,34)(12,21,26,38,25,36,27,37)(17,29,24,39,32,40,33,35);(3,5)(6,7)(9,12)(10,17)(11,21)(13,27)(14,26)(15,25)(16,29)(18,33)(19,24)(20,32)(22,38)(23,36)(28,37)(30,39)(31,40)(34,35)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1995);