Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 36864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#1009 |
| Fitting | C2×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D3 |
| commutator, cosocle | C2×C3 | C25 |
| Frattini | C2 | C24×D3 |
| Fitting | C2×C3×#41 | C2 |
| socle | C22×C3 | C24 |
| hypercenter | C2×#41 | D3 |
| hypocenter | C3 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 13 points:
(1,2)(3,4,5)(6,8,9,12)(7,11,10,13);(6,7)(8,11)(9,10)(12,13);(7,10)(11,13);(4,5)(8,12)(11,13);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | #70048, #70139, #70305, … |
| maximal normal subgroup of | #70048, #70139, #70305, … |
| maximal quotient of | C2×#5519, C2×#5521, C2×#5523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1521);