Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#7207 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #99 |
| commutator | C23 | C4×C8×C53 |
| Frattini, cosocle | C2×#131 | C22×C53 |
| socle | C2×C53 | #1282 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×C8×C53 ◁ C2×#131×C53 ◁ G | |
248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 117 points:
(1,2,4,10,6,12,20,33,7,9,18,28,23,30,43,52)(3,13,14,31,16,36,38,55,17,8,24,25,41,27,47,50)(5,11,19,32,21,34,44,56,22,29,42,48,45,53,60,62)(15,35,37,54,39,57,58,64,40,26,46,49,59,51,61,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,5,15,6,16,21,39,7,17,22,40,23,41,45,59)(2,8,11,26,12,27,34,51,9,13,29,35,30,36,53,57)(4,14,19,37,20,38,44,58,18,24,42,46,43,47,60,61)(10,31,32,54,33,55,56,64,28,25,48,49,52,50,62,63)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 195);