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 ⋊ #3191 |
| 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 | #3191 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #449 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(6,11)(8,10)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(2,3)(4,6)(5,8)(7,11)(9,10);(4,5)(6,10)(7,9)(8,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27)Polycyclic presentation:
2^7,3|d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,h^b=h^2,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3191, C2×#80090, #11665760, … |
| maximal normal subgroup of | D3×#3191, C2×#80090, #11665760, … |
| maximal quotient of | #10804204, #10804205, #10805449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12666);