Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C29×D3 (order 3072) · Inn(G) = C22×D3 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator | C22×C3 | C22×C4 |
| Fitting | C3×#95 | C2 |
| socle, prime radical, cosocle | C23×C3 | C23 |
| hypercenter | #95 | D3 |
| hypocenter | C3 | #389 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #95 | |
170 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,5,6,7)(9,11)(12,14,15,13);(8,9,10,11)(13,14);(2,3)(12,13,15,14)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=d,b^a=be,g^a=g^2,c^b=cf48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4647, #73334, #73337, … |
| maximal normal subgroup of | C2×#4647, #73334, #73337, … |
| maximal quotient of | #68617, #68618, #68747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 206);