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 | C2 | C2×#63419 |
| commutator, cosocle, hypocenter | #95⋊C9 | C22 |
| Frattini, socle | C23×C3 | C2×S4 |
| Fitting | C2×C3×#95 | D3 |
| hypercenter | C22 | #63419 |
| LCS | G ▷ #95⋊C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
253 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 21 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|c^3=d,e^2=g,f^2=ghi,b^a=bgh,c^a=c^2d^2,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,e^c=fh,f^c=efgh,g^c=ghi,h^c=i,i^c=hi,f^e=fgh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153951);