Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = #882 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#99 ⋊ C2 |
| Sylow | C3 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #882 |
| commutator, cosocle | C23×C3 | C23 |
| Frattini | C23 | C22×D3 |
| Fitting | C3×#99 | C2 |
| socle | C2×C3 | #120 |
| hypercenter | #99 | D3 |
| hypocenter | C3 | B(4,𝔽2) |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #99 | |
198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 198 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) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2,3)(4,5)(6,10,11,8)(7,9);(4,6)(5,8)(7,11)(9,10);(2,3)(4,5)(6,8)(7,9)(10,11)Polycyclic presentation:
2^6,3|c^2=d,c^a=cd,e^a=ef,g^a=g^2,c^b=ce,d^b=df,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | #69130, #69131, #69138, … |
| maximal normal subgroup of | #69130, #69131, #69138, … |
| maximal quotient of | #68106, #68107, #68108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 300);