Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#449 ⋊ C2 |
| Sylow | C3 ⋊ #3330 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#449 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #449 | D3 |
| hypocenter | C3 | #3330 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #449 | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(6,11)(8,10)(12,14,16,15)(13,17,18,19);(12,13)(14,17)(15,19)(16,18);(2,3)(4,6)(5,8)(7,11)(9,10)(13,18)(17,19);(2,3)(4,5,7,9)(6,10,11,8)(13,17)(14,15)(18,19)Polycyclic presentation:
2^7,3|a^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3330, C2×#80116, #11672136, … |
| maximal normal subgroup of | D3×#3330, C2×#80116, #11672136, … |
| maximal quotient of | #10810832, #10811608, #10812474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12692);