Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#7208 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #99 |
| commutator | C2×C4 | C4×C8×C53 |
| Frattini, cosocle | C2×#131 | C22×C53 |
| socle | C2×C53 | #1282 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ 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:
(1,2,5,11,7,10,24,34,8,13,26,38,4,20,21,36)(3,14,16,40,18,33,44,58,19,42,23,51,15,9,30,31)(6,27,25,54,28,49,53,61,29,12,39,37,22,35,50,60)(17,46,45,63,47,32,59,57,48,41,52,62,43,55,56,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,6,17,7,18,28,47,8,19,29,48,4,15,22,43)(2,9,12,32,10,14,35,41,13,33,27,55,20,42,49,46)(5,23,25,52,24,30,53,56,26,16,39,45,21,44,50,59)(11,31,37,57,34,40,60,62,38,58,54,64,36,51,61,63)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 196);