Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #64188⋊C2 |
| commutator, cosocle, hypocenter | #63703 | C22 |
| Frattini | C23×C3 | C2×S4 |
| Fitting | C3×B(4,𝔽2) | D3 |
| socle | C2×C3 | #1020⋊C2 |
| LCS | G ▷ #63703 | |
| UCS | C1 ◁ C2 | |
242 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 242 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) = 144 |
Polycyclic presentation:
2^2,3^2,2^5|c^3=d,b^a=bi,c^a=c^2d^2,d^a=d^2,e^a=ef,g^a=gh,e^b=egi,f^b=fhi,e^c=f,f^c=ef,g^c=h,h^c=gh,h^e=hi,g^f=gi33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153993);