Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#9038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #120 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| socle | C22×C73 | #386 |
| prime radical | #525×C73 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C23×C4×C73 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(18,22,25,28)(19,26,27,29)(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);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,18,23,22,21,25,20,28)(19,24,26,30,27,32,29,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,26)(25,32)(28,31)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 2026);