Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×S4 (order 2304) · Inn(G) = C22×#67 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#67 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×2−1+4 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | 2−1+4 | D3 |
| hypocenter | C3 | #577 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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)(4,6,8,13)(5,10,9,16)(7,15,14,12)(11,19,17,18);(5,9)(10,16)(11,17)(18,19);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,17)(15,19);(2,3)(6,12)(7,14)(10,18)(11,17)(13,15)(16,19)Polycyclic presentation:
2^6,3|d^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^2,c^b=cf,e^d=ef39 conjugacy classes — view the full character table.
| maximal subgroup of | #80287, #80289, #80301, … |
| maximal normal subgroup of | #80287, #80289, #80301, … |
| maximal quotient of | #82547, #82548, #82550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1396);