Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#1413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #354 |
| commutator | C42 | C2×C4×C7 |
| Frattini, cosocle | C4⋊Q8 | C22×C7 |
| prime radical | C22×C7 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×C4⋊Q8 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,16,13,18,25,35)(10,19,21,37,20,36,38,33)(12,17,27,34,28,15,24,32)(14,29,30,22,31,23,39,26);(8,10,13,20)(9,14)(11,23,25,29)(12,26,28,22)(15,19)(16,33)(17,36)(18,31)(21,24,38,27)(30,32)(34,39)(35,37)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 545);