Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C28 |
| direct product | C2×C73×#3121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C2×#121×C73 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×#117×C73 ◁ G | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 5 generators on 99 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(3,4)(5,8)(6,7)(9,10)(13,19)(14,21)(16,23)(18,25)(20,22)(24,26);(4,7)(8,10)(11,12)(13,16)(14,18)(15,17)(19,23)(20,24)(21,25)(22,26);(11,13,15,20)(12,16,17,24)(14,22,21,19)(18,26,25,23);(3,5,6,9)(4,8,7,10)(12,17)(16,24)(18,25)(23,26)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53538);