Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#8811 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C2×D4 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| socle, prime radical | C23×C73 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ C43×C73 ◁ G | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(23,29)(26,32)(30,35)(33,36)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1799);