Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C73 · Schur multiplier (H2) = C24×C4 |
| direct product | C73×#20510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C73 | C2×D4 |
| commutator | C4 | C22×C42×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | C2×C4×D4 |
| prime radical | C2×#135×C73 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(5,6,10,15,11,13,9,23)(7,16,18,12,19,27,24,25)(8,14,21,29,22,28,20,33)(17,30,31,26,32,36,34,35)(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);(1,2,4,3)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,25)(14,26)(15,27)(16,23)(20,34)(21,31)(22,32)(28,35)(29,36)(30,33);(1,3,4,2)(6,13)(12,25)(14,28)(15,23)(16,27)(26,35)(29,33)(30,36);(5,8,10,21,11,22,9,20)(6,14,15,29,13,28,23,33)(7,17,18,31,19,32,24,34)(12,26,27,36,25,35,16,30)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13498);