Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 9216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, prime radical, cosocle | C22×C3 | C24 |
| Fitting | C3×C4×Q8 | C2 |
| hypercenter | C4×Q8 | D3 |
| hypocenter | C3 | #557 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,11)(5,8,9,10)(12,15,16,14)(13,17,18,19);(12,14,16,15)(13,17,18,19);(12,13,16,18)(14,19,15,17);(2,3)(4,5,7,9)(6,10,11,8)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=f,d^2=ef,d^a=de,g^a=g^2,c^b=cf45 conjugacy classes — view the full character table.
| maximal subgroup of | #79624, #79627, #79633, … |
| maximal normal subgroup of | #79624, #79627, #79633, … |
| maximal quotient of | #73642, #73652, #73658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1128);