Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2355 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2355 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,18,19,23)(17,21,20,22);(8,11,12,10)(9,13,14,15);(2,3)(8,10,12,11)(9,13,14,15)(16,17,19,20)(18,22,23,21);(4,5,6,7)(8,9)(10,13)(11,15)(12,14)(17,20)(21,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=ef,c^2=f,d^2=e,b^a=be,c^a=cf,d^b=de,h^b=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #10689353, #10689357, #10689381, … |
| maximal normal subgroup of | #10689353, #10689357, #10689381, … |
| maximal quotient of | C4⋊C4×C4⋊Dic3, #10716629, #10716675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6216);