Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×#120 (order 1536) · Inn(G) = C22×D3 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D3 |
| commutator, socle | C22×C3 | C22×C4 |
| Fitting | C3×C22⋊C8 | C2 |
| prime radical | C23×D3 | C4 |
| cosocle | C2×C3×C4 | C23 |
| hypercenter | C22⋊C8 | D3 |
| hypocenter | C3 | #407 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C22⋊C8 | |
190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(8,9,10,13,11,12,14,15);(4,5)(6,7);(2,3)(9,12)(13,15)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,c^a=cf,g^a=g^2,c^b=ce48 conjugacy classes — view the full character table.
| maximal subgroup of | #68120, #68121, #68122, … |
| maximal normal subgroup of | #68120, #68121, #68122, … |
| maximal quotient of | #68102, #68103, #68120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 285);