Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ C4×#126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#388 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | C4×#126 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(4,6,7,5);(9,13)(12,15)(16,18,19,23)(17,21,20,22);(2,3)(8,10,11,14)(9,12,13,15)(17,20)(21,22);(2,3)(4,5,7,6)(8,9)(10,12)(11,13)(14,15)(16,17)(18,22)(19,20)(21,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=ef,c^2=e,d^2=g,b^a=be,c^a=cf,h^a=h^2,h^b=h^2,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×#126, C2×#75455, C4×#5649, … |
| maximal normal subgroup of | C4×D3×#126, C2×#75455, C4×#5649, … |
| maximal quotient of | Dic3×#388, #10716348, #10716359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8031);