Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#25409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(2,3)(5,6,10,14,9,13,16,26)(7,15,19,32,18,11,25,28)(8,20,23,30,22,31,24,12)(17,33,34,27,21,36,35,29)(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);(1,2,4,3)(5,7,16,25,9,18,10,19)(6,11,26,32,13,15,14,28)(8,21,24,34,22,17,23,35)(12,29,31,36,30,27,20,33);(5,8,9,22)(6,12,13,30)(7,17,18,21)(10,24,16,23)(11,27,15,29)(14,31,26,20)(19,35,25,34)(28,33,32,36);(1,3,4,2)(6,13)(11,15)(12,30)(14,26)(20,31)(27,29)(28,32)(33,36)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18397);