Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×C43 ⋊ C2 |
| Sylow | C3 ⋊ #2298 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator | C22×C3 | C23×C4 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×C43 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #2298 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
524 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 205 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)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,13,12,15)(17,20)(21,23);(4,6,7,5)(9,12)(13,15);(2,3)(4,5,7,6)(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=e,d^2=f,b^a=be,c^a=cf,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10688638, #10688642, #10793411, … |
| maximal normal subgroup of | #10688638, #10688642, #10793411, … |
| maximal quotient of | #10716307, #10716318, #10716329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5877);