Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#33527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C73 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#41×C73 ◁ C2×#131×C73 ◁ G | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,22)(10,20)(11,15)(12,18)(14,24)(17,33)(19,34)(21,31)(23,27)(25,32)(26,30)(28,36)(29,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7,11,19)(6,12,15,29)(8,17,10,26)(9,23,25,36)(13,27,20,28)(14,30,31,33)(16,32,34,22)(18,21,35,24);(2,4)(5,6)(7,12)(8,20)(9,14)(10,13)(11,15)(16,18)(17,28)(19,29)(21,32)(22,24)(23,30)(25,31)(26,27)(33,36)(34,35);(5,8,11,10)(6,13,15,20)(7,17,19,26)(9,21,25,24)(12,27,29,28)(14,22,31,32)(16,33,34,30)(18,23,35,36)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26515);