Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #442 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,16)(18,19)(20,30)(21,33,35,41)(22,36)(23,25)(24,29)(26,31,44,49)(27,38)(28,32)(34,46,37,47)(39,45,48,43)(40,42);(14,15,17,16)(18,20,22,23,24,32,38,40)(19,25,27,30,29,42,36,28)(21,31,37,39,35,49,34,48)(26,41,45,47,44,33,43,46);(14,16,17,15)(18,21,24,35)(19,26,29,44)(20,31,32,49)(22,37,38,34)(23,39,40,48)(25,41,42,33)(27,45,36,43)(28,46,30,47)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6129);