Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×D8 ⋊ C2 |
| Sylow | C3 ⋊ #3297 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3297 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
420 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)(6,11)(8,10);(13,18)(16,23)(17,24)(19,20)(22,27)(25,26);(2,3)(4,6)(5,8)(7,11)(9,10)(12,14,15,21)(13,17,19,25)(16,22,23,27)(18,24,20,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,19)(14,17,21,25)(16,18,23,20)(22,24,27,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,d^b=deg,h^b=h^2,f^c=fg,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3297, C2×#79609, #11671668, … |
| maximal normal subgroup of | D3×#3297, C2×#79609, #11671668, … |
| maximal quotient of | #10810609, #10811443, #10812366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12185);