Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#452 ⋊ C2 |
| Sylow | C3 ⋊ #3224 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#452 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #452 | D3 |
| hypocenter | C3 | #3224 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #452 | |
408 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)(13,19)(14,22)(18,20)(23,27);(2,3)(6,11)(8,10)(13,18)(14,22)(16,24)(17,25)(19,20)(21,26);(2,3)(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,22)(16,23)(18,26)(19,21)(20,25)(24,27);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,25)(26,27)Polycyclic presentation:
2^7,3|b^2=eg,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,d^b=dg,h^b=h^2,f^c=fg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3224, C2×#80263, #11666595, … |
| maximal normal subgroup of | D3×#3224, C2×#80263, #11666595, … |
| maximal quotient of | #10804559, #10804953, #10805793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12839);