Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#7252 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #99 |
| commutator | C22×C4 | C42×C73 |
| Frattini, cosocle | C2×#117 | C22×C73 |
| socle | C22×C73 | #342 |
| prime radical | C24×C73 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×C4×C73 ◁ C2×#117×C73 ◁ G | |
344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 344 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) = 113 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 2 generators on 113 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,26,33)(11,19,21,34,22,30,35,39)(12,16,24,31,25,32,37,40)(15,27,28,23,29,36,38,20)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5,7,6)(9,11)(10,15)(12,23,25,20)(13,21)(14,22)(16,30,32,19)(17,28)(18,29)(24,36,37,27)(26,35)(31,39,40,34)(33,38)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 240);