Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C73 · Schur multiplier (H2) = C22 |
| direct product | C73×#7196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #95 |
| commutator | C2×C4 | C4×C8×C73 |
| Frattini, cosocle | C2×C4×C8 | C22×C73 |
| socle | C22×C73 | #336 |
| prime radical | C23×C73 | #97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C4×C8×C73 ◁ G | |
210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 2 generators on 105 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,20,25,21,23,29,30,22,26,28,32,19,24,27,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(18,23,26,24)(20,28)(25,31,32,30)(27,29)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 184);