Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C2×H(5,𝔽2) ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C22 |
| Fitting | C2×H(5,𝔽2) | C3⋊D3 |
| hypercenter | C22 | A4⋊S4 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
378 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 378 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.
| 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,c^a=c^2,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,h^a=hi,g^c=h,h^c=ghi,e^d=f,f^d=efi,f^e=fi,h^g=hi37 conjugacy classes — view the full character table.
| maximal subgroup of | #72224794, C2×#11923518, #72228298, … |
| maximal normal subgroup of | #72224794, C2×#11923518, #72228298, … |
| maximal quotient of | #72226037, #72226061, #72226069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157529);