Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ #3257 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3257 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,21)(17,25)(22,27)(24,26);(5,8)(9,11)(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(2,3)(4,5)(6,9)(7,8)(10,11)(12,14)(13,17)(15,22)(16,21)(18,24)(19,26)(20,25)(23,27);(5,8)(6,10)(12,13)(14,17)(15,18)(16,20)(19,23)(21,25)(22,24)(26,27)Polycyclic presentation:
2^7,3|d^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,d^b=dg,h^b=h^2,f^c=fg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3257, C2×#80346, #11668170, … |
| maximal normal subgroup of | D3×#3257, C2×#80346, #11668170, … |
| maximal quotient of | #10805157, #10805173, #10805595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12922);