Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#12748 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #447 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C2×#576×C73 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C22×C4×C73 ◁ C2×#131×C73 ◁ G | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,4)(5,6,9,15,11,16,10,27)(7,17,19,32,20,31,26,13)(8,21,23,33,25,35,24,14)(12,22,28,34,30,36,29,18)(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)(6,12)(8,22)(9,19)(10,26)(11,20)(13,14)(15,28)(16,30)(17,21)(18,24)(23,34)(25,36)(27,29)(31,35)(32,33);(2,4)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,14)(15,31)(16,32)(17,27)(21,29)(22,24)(23,36)(25,34)(28,35)(30,33)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5736);