Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2932 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting, prime radical | C3×#532 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #2932 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,11)(9,14)(10,16)(12,20)(13,19)(15,18)(17,23)(21,22);(2,3)(8,10,12,15,13,17,21,9)(11,16,20,18,19,23,22,14);(5,7)(11,19)(14,18)(16,23)(20,22);(2,3)(4,5)(6,7)(8,9,12,10,13,15,21,17)(11,18,20,23,19,14,22,16)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,c^a=cg,d^a=dg,h^a=h^2,d^b=dg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10689317, #10689321, #10689331, … |
| maximal normal subgroup of | #10689317, #10689321, #10689331, … |
| maximal quotient of | #10765895, #10765903, #10765910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15852);