Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2378 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#380 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2378 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 464 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(9,13)(10,14)(17,20)(21,23);(2,3)(8,10)(9,12)(11,14)(13,15)(16,17,19,20)(18,23,22,21);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(17,20)(18,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,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 | #10692214, #10692218, #10692228, … |
| maximal normal subgroup of | #10692214, #10692218, #10692228, … |
| maximal quotient of | #10716808, #10716820, #10716830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6462);