Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72217027 (order 9216) · Inn(G) = C23×D3 · Out(G) = C23×S4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×D3 |
| commutator, socle | C2×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#41 | C2 |
| cosocle | C3×C4 | C24 |
| hypercenter | C2×#41 | D3 |
| hypocenter | C3 | #568 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,10)(9,15)(11,17)(16,19);(6,12)(9,15)(13,18)(16,19);(4,6)(5,9)(7,13)(8,12)(10,15)(11,16)(14,18)(17,19);(2,3)(4,5,7,11,8,10,14,17)(6,9,13,16,12,15,18,19)Polycyclic presentation:
2^6,3|a^2=e,e^2=f,d^a=df,g^a=g^2,c^b=cf54 conjugacy classes — view the full character table.
| maximal subgroup of | #70001, #70002, #70044, … |
| maximal normal subgroup of | #70001, #70002, #70044, … |
| maximal quotient of | #82202, #82204, #82209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1378);