Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#12758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #1790 |
| commutator | C2×D4 | C22×C4×C73 |
| Frattini, cosocle | C2×#41 | C23×C73 |
| prime radical | C73×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C2×C4×C73 ◁ C2×#41×C73 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,6,11,7,12,4,9)(3,13,17,28,16,29,14,30)(5,19,18,26,20,10,21,25)(8,22,23,15,24,31,27,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,7,16)(2,8,12,24)(4,14,6,17)(5,15,20,32)(9,23,11,27)(10,13,19,29)(18,31,21,22)(25,30,26,28);(1,4,7,6)(2,9,12,11)(3,14,16,17)(5,18,20,21)(8,23,24,27)(10,26,19,25)(13,28,29,30)(15,31,32,22)2482 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5746);