Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#17856 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C22×D4 |
| commutator | C4 | C23×C8×C73 |
| Frattini, cosocle | C42 | C24×C73 |
| socle | C22×C73 | C2×C4×D4 |
| prime radical | C23×C73 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ #431×C73 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2,4,7,5,3,6,8)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,31)(24,37)(26,29)(28,35)(30,36)(34,40)(38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,31,39)(18,32,25,19)(24,38,36,35)(30,40,37,33);(1,3,4,8,5,2,6,7)(9,12,14,23)(10,16,17,31)(11,20,22,34)(13,24,27,36)(15,28,29,39)(18,30,25,37)(19,33,32,40)(21,35,26,38);(10,17)(12,23)(15,29)(18,25)(19,32)(20,34)(24,36)(35,38)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 10844);