Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#431 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C22 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(7,10)(11,13)(14,15,16,18,17,19,20,21);(6,7)(8,11)(9,10)(12,13);(1,2)(6,8,9,12)(7,11,10,13)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17357);