Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S42 |
| commutator, prime radical, cosocle | C2×A42 | C22 |
| Fitting, socle | C25 | D32 |
| hypocenter | A42 | Q8 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2 | |
383 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 383 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.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^2,2^4|a^2=c,b^2=c,b^a=bc,d^a=d^2,h^a=hi,e^b=e^2,f^b=fg,h^d=i,i^d=hi,f^e=g,g^e=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #72225965, #72228048, #72228053, … |
| maximal normal subgroup of | #72225965, #72228048, #72228053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157472);