Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #518 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator | C2×C3 | C23×C4 |
| Fitting | C3×#117 | C2 |
| socle, cosocle | C22×C3 | C24 |
| prime radical | C22×D12 | C2 |
| hypercenter | #117 | D3 |
| hypocenter | C3 | #518 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(4,5,6,7)(9,13)(12,15);(8,10,11,14)(9,12,13,15);(8,9)(10,12)(11,13)(14,15);(2,3)(10,14)(12,15)Polycyclic presentation:
2^6,3|c^2=f,d^2=e,c^a=cf,g^a=g^2,d^b=df54 conjugacy classes — view the full character table.
| maximal subgroup of | #68704, #68705, #68758, … |
| maximal normal subgroup of | #68704, #68705, #68758, … |
| maximal quotient of | #73389, #73391, #73395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1080);