Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#440 ⋊ C2 |
| Sylow | C3 ⋊ #3309 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#440 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #440 | D3 |
| hypocenter | C3 | #3309 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #440 | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(6,11)(8,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(12,14,17,22)(13,18,20,26)(15,23,16,25)(19,27,24,21);(2,3)(4,6)(5,8)(7,11)(9,10)(14,22)(18,26)(21,27)(23,25);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,21)(15,19)(16,24)(17,20)(18,23)(22,27)(25,26)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,f^c=fg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3309, C2×#79799, #11671970, … |
| maximal normal subgroup of | D3×#3309, C2×#79799, #11671970, … |
| maximal quotient of | #10810717, #10811585, #10812311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12375);