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 ⋊ #2320 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #2320 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
444 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(16,19,20,18)(17,22,21,23);(9,12)(13,15)(16,18,20,19)(17,22,21,23);(16,17,20,21)(18,23,19,22);(2,3)(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(17,21)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=e,d^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=ce84 conjugacy classes — view the full character table.
| maximal subgroup of | #10692115, #10692119, #10692129, … |
| maximal normal subgroup of | #10692115, #10692119, #10692129, … |
| maximal quotient of | #10716463, #10716480, #10716503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15030);