Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#123 ⋊ C2 |
| Sylow | C3 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#123 | C2 |
| prime radical | C2×#237 | C2 |
| hypercenter | #123 | D3 |
| hypocenter | C3 | #536 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #123 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(8,11,12,19)(9,14,15,18)(10,17,16,22)(13,21,20,23);(4,5)(6,7)(10,16)(13,20)(17,22)(21,23);(8,10)(9,13)(11,17)(12,16)(14,21)(15,20)(18,23)(19,22);(2,3)(8,9,12,15)(10,13,16,20)(11,18,19,14)(17,23,22,21)Polycyclic presentation:
2^6,3|a^2=f,d^2=f,d^a=df,g^a=g^2,c^b=cf,d^c=de36 conjugacy classes — view the full character table.
| maximal subgroup of | #69145, #80309, C2×#5648, … |
| maximal normal subgroup of | #69145, #80309, C2×#5648, … |
| maximal quotient of | #74053, #74056, #74064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1207);