Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C22×#508×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | D8 |
| commutator, Frattini, cosocle | C8 | C25×C73 |
| socle | C23×C73 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ C23×C4×C73 ◁ C23×C8×C73 ◁ G | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2)(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);(3,4);(5,6)(7,16)(8,13)(9,21)(10,25)(11,15)(12,18)(14,23)(17,32)(19,33)(20,31)(22,27)(24,35)(26,30)(28,36)(29,34);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,30)(15,29)(16,21)(18,23)(20,34)(24,36)(25,28)(31,32)(33,35);(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,27,29,28)(14,21,31,35)(16,32,33,30)(18,22,34,36)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55616);