Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#381 ⋊ C2 |
| Sylow | C3 ⋊ #2308 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#381 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D12 | C2 |
| hypercenter | #381 | D3 |
| hypocenter | C3 | #2308 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #381 | |
1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 1160 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) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(13,17)(16,18);(2,3)(9,11)(14,19)(16,18);(2,3)(12,14)(13,16)(15,19)(17,18);(4,5,6,7)(8,9)(10,11)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|a^2=g,b^a=be,c^a=cf,d^a=de,c^b=ce,h^b=h^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10688702, #10688705, #10692903, … |
| maximal normal subgroup of | #10688702, #10688705, #10692903, … |
| maximal quotient of | #10716427, #10716431, #10716483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6695);