Finite Group DatabasebetaFinite Group Databasebeta| order | 4672 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C4⋊C4×C73 |
| unambiguous semidirect product | C4⋊(C42×C73) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C22 |
| commutator | C2 | C2×C42×C73 |
| Frattini, cosocle | C23 | C23×C73 |
| socle, prime radical | C23×C73 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ G | |
194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,3,4)(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);(6,8)(9,10,11,12);(5,6,7,8)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4672, 59);