Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#121 ⋊ C2 |
| Sylow | C3 ⋊ #449 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#67 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×#121 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | #121 | D3 |
| hypocenter | C3 | #449 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #121 | |
162 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(9,14)(10,11)(13,15);(4,5)(6,7)(8,10,12,11)(9,13,15,14);(2,3)(8,9)(10,13)(11,14)(12,15)Polycyclic presentation:
2^6,3|b^2=f,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #79998, #79999, #80004, … |
| maximal normal subgroup of | #79998, #79999, #80004, … |
| maximal quotient of | #69506, #69522, #69524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 596);