Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C22×D3 · Out(G) = C26 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Frattini | C22 | C23×D3 |
| Fitting | C3×C4×D4 | C2 |
| prime radical | #939 | C2 |
| hypercenter | C4×D4 | D3 |
| hypocenter | C3 | #529 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
234 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,9)(6,10)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(4,6)(5,8)(7,10)(9,11);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,16,18,19)Polycyclic presentation:
2^6,3|a^2=f,c^2=e,d^a=df,g^a=g^2,d^b=def48 conjugacy classes — view the full character table.
| maximal subgroup of | #79400, #79402, #79407, … |
| maximal normal subgroup of | #79400, #79402, #79407, … |
| maximal quotient of | #73437, #73438, #73439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1097);