Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×Q8×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C22 |
| commutator | C2 | C23×C4×C53 |
| Frattini, cosocle | C22 | C24×C53 |
| socle, prime radical | C23×C53 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2)(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);(3,4,5,6);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 197);