Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#33568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C73 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C23×C73 ◁ C22×Q8×C73 ◁ G | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2)(3,12)(4,16)(5,8)(6,11)(7,14)(9,19)(10,21)(13,27)(15,29)(17,24)(18,26)(20,30)(22,28)(23,31)(25,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3)(2,7)(4,17)(5,20)(6,13)(8,23)(9,27)(10,25)(11,24)(12,16)(14,19)(15,21)(18,31)(22,32)(26,29)(28,30);(2,8)(5,21)(7,23)(9,28)(10,11)(12,29)(15,20)(16,26)(18,22)(24,25)(27,30)(31,32);(3,13)(7,24)(12,27)(14,17)(15,20)(23,25)(29,30)(31,32)2482 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26556);