Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C27 |
| direct product | C73×#23446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C73 |
| prime radical | C73×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,22)(16,23)(17,24)(18,26)(19,27)(20,28)(21,29)(25,31)(30,34)(32,35)(33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7,10,17)(6,11,14,24)(8,15,19,31)(9,20,21,16)(12,25,27,22)(13,28,29,23)(18,32,33,30)(26,34,36,35);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,26)(14,27)(16,30)(17,31)(20,32)(21,33)(23,34)(24,25)(28,35)(29,36);(5,9,10,21)(6,13,14,29)(7,16,17,20)(8,18,19,33)(11,23,24,28)(12,26,27,36)(15,30,31,32)(22,34,25,35)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 16434);