Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C73 · Schur multiplier (H2) = C213 |
| direct product | C2×Q8×#41×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C73 |
| socle | C23×C73 | C25 |
| prime radical | C22×#41×C73 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ G | |
7522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 6 generators on 91 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(3,4,6,8)(5,9,10,7);(3,5,6,10)(4,7,8,9);(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)7300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55701);