Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #953 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #953 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×2−1+4 | C2 |
| socle | C2×C3 | #120 |
| hypercenter | 2−1+4 | D3 |
| hypocenter | C3 | #454 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ 2−1+4 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,8,13)(5,10,11,19)(7,16,14,12)(9,17,15,18);(4,5)(6,10)(7,15)(8,11)(9,14)(12,17)(13,19)(16,18);(2,3)(5,9)(7,14)(10,18)(11,15)(12,16)(17,19)Polycyclic presentation:
2^6,3|c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,d^b=df,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #70987, #70988, #70993, … |
| maximal normal subgroup of | #70987, #70988, #70993, … |
| maximal quotient of | #71658, #71661, #71665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 804);