Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#20776 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | C2×#117 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C2×C73 | C2×#383 |
| prime radical | C24×C73 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C22×M16×C73 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2)(3,12)(4,16,18,23)(5,10)(6,11)(7,15)(8,17,24,30)(9,19)(13,20,26,31)(14,22)(21,28)(25,27,29,32)(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);(1,3,6,14)(2,7)(4,13,18,26)(5,15,19,28)(8,20)(9,12)(10,22)(11,21)(16,29)(17,27,30,32)(23,25)(24,31);(1,4,5,17,6,18,19,30)(2,8,9,23,11,24,10,16)(3,13,15,27,14,26,28,32)(7,20,12,25,21,31,22,29);(3,14)(7,21)(12,22)(13,26)(15,28)(20,31)(25,29)(27,32)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13764);