Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#13228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#99 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C22×Q8 | C23×C73 |
| prime radical | C22×D4×C73 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×C4×Q8×C73 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 93 points:
(2,4)(5,6,10,14,11,15,8,12)(7,16,18,9,13,17,20,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(5,7,17,14,11,13,16,12)(6,10,20,9,15,8,18,19);(1,2,3,4)(5,8,11,10)(6,12,15,14)(7,18,13,20)(9,17,19,16)2701 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6216);