Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | (#1020⋊C3)⋊C2 |
| commutator, prime radical, cosocle, hypocenter | Ω+(4,𝔽3) | C22 |
| Fitting | H(5,𝔽2) | D32 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 | |
141 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=i,b^2=i,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,c^b=cd^2,d^b=d^2,e^b=gh,f^b=hi,g^b=ef,h^b=fi,e^c=f,f^c=efi,e^d=f,f^d=efi,g^d=h,h^d=ghi,f^e=fi,h^g=hi22 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157477);