Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#24470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(5,6,11,16,12,17,10,26)(7,18,9,13,20,15,25,29)(8,21,23,32,24,31,27,14)(19,34,35,30,36,28,22,33)(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)(3,4)(5,7)(6,13)(8,19)(9,11)(10,25)(12,20)(14,28)(15,16)(17,29)(18,26)(21,33)(22,27)(23,35)(24,36)(30,31)(32,34);(2,4)(5,8,12,24)(6,14,17,32)(7,19,20,36)(9,22,25,35)(10,23,11,27)(13,28,29,34)(15,30,18,33)(16,31,26,21);(5,9)(6,15)(7,10)(8,22)(11,20)(12,25)(13,26)(14,30)(16,29)(17,18)(19,23)(21,28)(24,35)(27,36)(31,34)(32,33)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 17458);