Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C71 · Schur multiplier (H2) = C27 |
| direct product | C2×C71×#2015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C71 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C71 |
| prime radical | C23×D4×C71 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C71 ◁ C22×C42×C71 ◁ G | |
2166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 9088 |
A minimal faithful permutation representation — 4 generators on 97 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);(3,4)(5,6)(11,12,16,20)(13,21,24,25)(14,19,15,18)(17,23,26,22);(4,6)(7,8)(9,10)(11,13,16,24)(12,17,20,26)(14,23,15,22)(18,25,19,21);(8,10)(12,18)(14,15)(17,25)(19,20)(21,26)(22,23)4544 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 16406);