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×#131×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C22 |
| commutator | C2 | C23×C4×C53 |
| Frattini, cosocle | C4 | C24×C53 |
| socle | C22×C53 | C22×C4 |
| prime radical | C2×#41×C53 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 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);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 248);