Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | #95⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10588306 |
| commutator, hypocenter | #95⋊C9 | C4 |
| Frattini | C22×C3×C4 | S4 |
| Fitting, prime radical | C3×#392 | D3 |
| socle | C23×C3 | SL(2,ℤ/4) |
| cosocle | #392⋊C9 | C2 |
| LCS | G ▷ #95⋊C9 | |
| UCS | C1 ◁ C2 | |
131 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=b,b^2=gh,c^3=d,e^2=g,f^2=ghi,c^a=c^2d^2,d^a=d^2,e^a=efh,f^a=fg,g^a=gi,h^a=hi,e^b=eh,f^b=fi,e^c=fhi,f^c=efg,g^c=ghi,h^c=i,i^c=hi,f^e=fgh39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153939);