Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #882 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#104 ⋊ C2 |
| Sylow | C3 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #882 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×#104 | C2 |
| socle | C2×C3 | #120 |
| hypercenter | #104 | D3 |
| hypocenter | C3 | #454 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #104 | |
146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 3 generators on 19 points:
(1,2,3)(4,7,18,6,9,15,8,16)(5,11,13,19,14,10,12,17);(4,6,9,16)(5,10,14,11)(7,18,15,8)(12,19,13,17);(2,3)(4,5)(6,10)(7,17)(8,13)(9,14)(11,16)(12,18)(15,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=df,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,e^b=ef,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #78986, C2×#4827, #78994, … |
| maximal normal subgroup of | #78986, C2×#4827, #78994, … |
| maximal quotient of | #68180, #68181, #68187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 386);