Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C73×#2244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C73 |
| prime radical | C2×#493×C73 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 99 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(3,4)(5,8)(6,7)(9,10)(12,18)(13,24)(14,19)(15,26)(16,20)(21,22)(23,25);(4,7)(8,10)(11,12,19,13,16,22,26,25,17,23,15,21,20,24,14,18);(3,5,6,9)(4,8,7,10)(11,13,14,12,20,18,15,24,17,21,26,23,16,25,19,22)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26687);