Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C32×#120 ⋊ C22 |
| Sylow | C32 ⋊ #3025 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10595278 |
| commutator, cosocle | C23×C32 | C24 |
| Frattini | C23 | C22×D32 |
| Fitting | C32×#120 | C22 |
| socle | C2×C32 | C2×#120 |
| hypercenter | #120 | D32 |
| hypocenter | C32 | #3025 |
| LCS | G ▷ C23×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
1697 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1697 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^7,3^2|d^2=g,b^a=be,c^a=cf,f^b=fg,e^c=eg,i^c=hi^2,h^d=h^2,i^d=i^278 conjugacy classes — view the full character table.
| maximal subgroup of | #72214743 |
| maximal normal subgroup of | #72214743 |
| maximal quotient of | #72260356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 99322);