Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | #63703 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #64188 |
| commutator, hypocenter | #63703 | C4 |
| Frattini, socle | C22×C3 | #1020 |
| Fitting, prime radical | C2×C3×H(5,𝔽2) | D3 |
| cosocle | C2×#63703 | C2 |
| LCS | G ▷ #63703 | |
| UCS | C1 ◁ C22 | |
425 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=b,c^3=d,c^a=c^2d^2,d^a=d^2,e^a=ef,g^a=gh,e^c=f,f^c=ef,g^c=h,h^c=gh,h^e=hi,g^f=gi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154474);