Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C2×2−1+4×C53 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C53 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 5 generators on 71 points:
(1,2)(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);(4,8)(6,14)(9,11)(12,18);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 265);