Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C73 · Schur multiplier (H2) = C4 |
| direct product | C73×#7134 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | #333 |
| commutator | C42 | C2×C8×C73 |
| Frattini, cosocle | C43 | C22×C73 |
| socle | C2×C73 | #1272 |
| prime radical | C23×C73 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C22×C4×C73 ◁ C43×C73 ◁ G | |
230 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 1168 |
A minimal faithful permutation representation — 2 generators on 89 points:
(1,2)(3,10,12,6,11,15,13,16)(4,7)(5,9)(8,14)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,3,5,11)(2,6)(4,13,14,12)(7,10)(8,15)(9,16)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 122);