Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #985⋊A4 |
| unambiguous semidirect product | #87497⋊C3 |
| Fitting | #585 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×S4 |
| commutator, cosocle, hypocenter | #994 | C22×C3 |
| Frattini, socle | C2 | C2×A4×S4 |
| Fitting | #585 | C3×D3 |
| LCS | G ▷ #994 | |
| UCS | C1 ◁ C4 | |
380 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^2,3,2^2,3,2^3|b^2=i,d^2=i,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,d^c=de,e^c=d,e^d=ei,g^f=h,h^f=ghi,h^g=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72226048, #72226326, #72226329, … |
| maximal normal subgroup of | #72226048, #72226326, #72226329, … |
| maximal quotient of | GL(2,𝔽3)×#41⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157503);