Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×D4×AGL(1,ℤ/8) (order 1536) · Inn(G) = C2×D3×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×D8 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | C2×D8 | D3 |
| hypocenter | C3 | #576 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(6,12)(7,15)(10,18)(11,17)(13,14)(16,19);(4,6)(5,10)(7,12)(8,14)(9,16)(11,18)(13,15)(17,19);(5,9)(10,16)(11,17)(18,19);(2,3)(4,5)(6,10)(7,11)(8,9)(12,18)(13,19)(14,16)(15,17)Polycyclic presentation:
2^6,3|e^2=f,b^a=bf,g^a=g^2,d^c=de,e^c=ef,e^d=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69258, #69259, #71471, … |
| maximal normal subgroup of | #69258, #69259, #71471, … |
| maximal quotient of | #78726, #78731, #78737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1316);