Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2596 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#388 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2596 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
398 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)(5,9)(8,10)(13,16)(17,18)(20,22,23,27)(21,25,24,26);(5,9)(8,10)(12,14,15,19)(13,17,16,18);(2,3)(4,6,7,11)(5,8,9,10)(13,16)(14,19)(21,24)(25,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,16)(14,18,19,17)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=g,d^2=efg,b^a=be,c^a=cfg,d^a=de,h^a=h^2,c^b=cg,h^b=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2596, C2×#75636, #10965788, … |
| maximal normal subgroup of | D3×#2596, C2×#75636, #10965788, … |
| maximal quotient of | #10717679, #10717830, #10718784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8212);