Finite Group DatabasebetaFinite Group Databasebeta| order | 4672 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22 |
| direct product | #440×C73 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×D4 |
| commutator | C4 | C22×C4×C73 |
| Frattini, cosocle | C2×C4 | C23×C73 |
| prime radical | C2×D4×C73 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C42×C73 ◁ G | |
120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(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)(7,12)(10,11);(1,3,4,2)(6,10)(9,11)1606 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4672, 121);