Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C71 · Schur multiplier (H2) = C25 |
| direct product | C71×#23735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C71 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C71 |
| prime radical | C23×D4×C71 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C71 ◁ C2×#117×C71 ◁ G | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 4 generators on 99 points:
(5,6,8,10)(7,11,12,9)(13,14,16,20,18,22,17,21)(15,23,25,28,24,27,26,19)(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);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,23)(22,28);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(15,24)(19,28)(23,27)(25,26)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 16723);