Finite Group DatabasebetaFinite Group Databasebeta| order | 4672 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | #518×C73 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C24 |
| commutator | C2 | C23×C4×C73 |
| Frattini, cosocle | C22 | C24×C73 |
| prime radical | C22×D4×C73 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1168 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(6,9)(10,11);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)2482 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4672, 199);