Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C191 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, prime radical, cosocle | C22×C191 | C24 |
| Fitting | C4×Q8×C191 | C2 |
| hypercenter | C4×Q8 | D191 |
| hypocenter | C191 | #557 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
200 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,191|a^2=f,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,g^a=g^190,d^b=de1925 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37571615, #37571619, #37571677, … |
| maximal normal subgroup of | #37571615, #37571619, #37571677, … |
| maximal quotient of | Q8×C4⋊Dic191, #37575438, #37575475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1278);