Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #10588305⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10588305⋊C2 |
| unambiguous semidirect product | (#64188⋊C2)⋊C2 |
| unambiguous semidirect product | #10589552⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #120⋊C9 | C22 |
| Frattini | C23×C3 | C2×S4 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | GL(2,ℤ/4) |
| hypocenter | C22⋊(C22⋊C9) | D4 |
| LCS | G ▷ #120⋊C9 ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
220 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^2,2^4|d^3=e,b^a=bc,d^a=d^2e^2,e^a=e^2,f^a=fg,h^a=hi,f^b=fi,g^b=gh,f^c=fhi,g^c=gi,f^d=g,g^d=fg,h^d=hi,i^d=h33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154068);