Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×#135×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C53 |
| socle | C22×C53 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)1484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 253);