Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #4625⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#4625⋊C2 |
| unambiguous semidirect product | #4625⋊(C2×C3) |
| unambiguous semidirect product | (#1020⋊C2)⋊(C2×C3) |
| Fitting | C3×H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #4625⋊C2 |
| commutator, cosocle | #120⋊C3 | C22×C3 |
| Frattini | C23 | C2×C3×S4 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | GL(2,ℤ/4) |
| hypocenter | C22⋊A4 | C3×D4 |
| LCS | G ▷ #120⋊C3 ▷ C22⋊A4 | |
| UCS | C1 ◁ C3 | |
262 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(2,ℤ/4), C3×#11921467, D3×#4625⋊C2, … |
| maximal normal subgroup of | D3×#4625⋊C2, C2×C3×#4625⋊C2 |
| maximal quotient of | C2×C3×#4625⋊C2, C3×#72215875 |
| automorphism group of | #120⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154891);