Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12939 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×D8 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C4×D8 |
| prime radical | C2×#135×C73 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
398 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)(5,6,11,17,12,19,30,18)(7,20,24,36,26,14,25,35)(8,27,22,13,28,33,23,34)(9,29,21,15,10,31,32,16)(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);(2,4)(5,7,11,24,12,26,30,25)(6,13,17,33,19,34,18,27)(8,21,22,10,28,32,23,9)(14,16,35,29,20,15,36,31);(5,8)(6,14)(7,21)(9,25)(10,24)(11,22)(12,28)(13,16)(15,34)(17,35)(18,36)(19,20)(23,30)(26,32)(27,31)(29,33)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5927);