Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Frattini | C22 | C23×D3 |
| Fitting | C2×C3×C42 | C2 |
| prime radical | C23×C3 | C23 |
| hypercenter | C2×C42 | D3 |
| hypocenter | C3 | #533 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(4,6,7,11)(5,8,9,10);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=e,b^a=bf,d^a=de,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #68779, #69396, #69433, … |
| maximal normal subgroup of | #68779, #69396, #69433, … |
| maximal quotient of | #73300, C4×#870, C4×#869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1029);