Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#11018 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #120 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| socle | C22×C73 | #386 |
| prime radical | C2×#575×C73 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C23×C4×C73 ◁ G | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,4)(5,6,10,14,11,16,9,15)(7,17,19,12,20,27,25,28)(8,13,23,30,24,31,22,32)(18,29,33,26,34,36,21,35)(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)(6,12)(8,21)(9,25)(10,19)(11,20)(13,29)(14,27)(15,17)(16,28)(18,23)(22,34)(24,33)(26,30)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,33)(20,34)(21,25)(27,36)(28,35)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 4006);