Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (2−1+4⋊C3)⋊A4 |
| unambiguous semidirect product | #87536⋊C3 |
| unambiguous semidirect product | #87547⋊C3 |
| Fitting, Sylow | H(7,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×A42 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C32 |
| Fitting | H(7,𝔽2) | C32 |
| hypercenter | D4 | A42 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
625 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3^2,2^5|e^2=i,f^2=i,g^2=i,h^2=i,b^a=bi,g^c=gh,h^c=g,e^d=ef,f^d=e,f^e=fi,h^g=hi73 conjugacy classes — view the full character table.
| maximal subgroup of | #72224794, C2×(2−1+4⋊C3)⋊A4, #72226022, … |
| maximal normal subgroup of | #72224794, C2×(2−1+4⋊C3)⋊A4, #72226022, … |
| maximal quotient of | C2×(2−1+4⋊C3)⋊A4, #72226033, D4×Ω+(4,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157683);