Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2382 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#382 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2382 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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)(16,19,20,23)(17,21,18,22);(9,12)(13,15)(16,18,20,17)(19,22,23,21);(2,3)(8,9)(10,15)(11,12)(13,14)(16,17,20,18)(19,22,23,21);(4,5,6,7)(9,12)(10,14)(17,18)(21,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10689552, #10689554, #10689556, … |
| maximal normal subgroup of | #10689552, #10689554, #10689556, … |
| maximal quotient of | #10716811, #10716830, #10716835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6737);