Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C4⋊C8×C113 ⋊ C2 |
| Sylow | C113 ⋊ #424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D113 |
| commutator, socle | C22×C113 | C22×C4 |
| Fitting | C4⋊C8×C113 | C2 |
| hypercenter | C4⋊C8 | D113 |
| hypocenter | C113 | #424 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
114 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,113|a^2=e,b^2=d,e^2=f,b^a=bd,c^b=cf,g^c=g^1121148 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #424×D113, C2×#20298623, #28399135, … |
| maximal normal subgroup of | #424×D113, C2×#20298623, #28399135, … |
| maximal quotient of | #28389343, #28389347, #28389351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 574);