Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #85874⋊C3 |
| Fitting | C3×#519 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×A4 |
| commutator, cosocle | C2×C3×Q8 | C23×C3 |
| Fitting | C3×#519 | C2×C3 |
| socle | C22×C3 | C23×A4 |
| hypercenter | C22⋊C4 | D3×A4 |
| hypocenter | C3×Q8 | C2×C3×D4 |
| LCS | G ▷ C2×C3×Q8 ▷ C3×Q8 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
757 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 757 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^3,3,2^3,3,2|c^2=ei,f^2=i,g^2=i,b^a=bi,c^a=ce,h^a=h^2,c^b=ci,f^d=fg,g^d=f,g^f=gi90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157228);