Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#12682 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×C22⋊C4 |
| commutator | C23 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×C22⋊C8 |
| prime radical | C22×#41×C73 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C23×C8×C73 ◁ G | |
930 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)(5,6,8,13,9,14,20,28,10,12,21,25,22,26,33,35)(7,15,17,27,18,29,31,36,16,11,19,23,30,24,32,34)(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)(6,11)(8,19)(9,18)(10,16)(12,15)(13,27)(14,24)(17,21)(20,32)(22,30)(23,25)(26,29)(28,36)(31,33)(34,35);(2,4)(7,16)(11,15)(17,19)(18,30)(23,27)(24,29)(31,32)(34,36)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5670);