Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊#85556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C3⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C3×#5465 | C2 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | #1020 |
| LCS | G ▷ C3×#5465 | |
| UCS | C1 ◁ C22 | |
260 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2,3^2,2^6|a^2=i,d^2=i,e^2=i,f^2=hi,g^2=hi,b^a=b^2,c^a=c^2,d^a=de,e^a=ei,f^a=fg,g^a=ghi,d^b=e,e^b=dei,f^b=g,g^b=fghi,e^d=ei,g^d=gh,f^e=fh,g^f=ghi36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157137);