Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #2361 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#121 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #2361 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(12,14,15,13);(9,11);(2,3)(4,6,7,5)(13,14);(2,3)(4,5,7,6)(8,9)(10,11)(12,13,15,14)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10815456, #10815463, #10815501, … |
| maximal normal subgroup of | #10815456, #10815463, #10815501, … |
| maximal quotient of | #10716606, #10716687, #10716756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15212);