Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C4⋊C4×C191 ⋊ C2 |
| Sylow | C191 ⋊ #537 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | C2×C4⋊C4×C191 | C2 |
| hypercenter | C2×C4⋊C4 | D191 |
| hypocenter | C191 | #537 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
294 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,191|d^2=e,b^a=be,c^a=cf,d^a=de,g^a=g^190,c^b=ce,g^c=g^1901922 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37571454, #37571457, #37571488, … |
| maximal normal subgroup of | #37571454, #37571457, #37571488, … |
| maximal quotient of | #37575113, #37575138, #37575203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1130);