Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×C22⋊C8 |
| prime radical | #532×C73 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C4×C8×C73 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,29,37,15,21,28,38,13,30,12,18)(11,22,25,36,24,17,23,33,26,35,39,32,31,34,27,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5)(3,7)(9,11)(10,17)(12,27)(13,31)(14,25)(15,26)(16,24)(18,36)(19,35)(20,33)(21,34)(22,30)(23,29)(28,39)(32,37)(38,40);(9,12,13,28,15,29,16,14)(10,18,30,38,21,37,19,20)(11,23,31,25,26,27,24,39)(17,32,22,33,34,36,35,40)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 4480);