Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#511819 (order 3072) · Inn(G) = C2×D12 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D12 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×M16 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | C2×M16 | D3 |
| hypocenter | C3 | #575 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×M16 | |
262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 262 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)(4,7,8,16,9,15,6,13)(5,11,10,14,12,18,17,19);(4,6,9,8)(5,10,12,17)(7,13,15,16)(11,14,18,19);(7,15)(11,18)(13,16)(14,19);(2,3)(4,5)(6,10)(7,14)(8,17)(9,12)(11,16)(13,18)(15,19)Polycyclic presentation:
2^6,3|c^2=f,d^2=e,e^2=f,d^a=de,e^a=ef,g^a=g^2,d^b=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #68698, #68703, #68719, … |
| maximal normal subgroup of | #68698, #68703, #68719, … |
| maximal quotient of | #78479, #78494, #78497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1307);