Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator | C22×C3 | C22×C4 |
| Fitting | C22×C3×Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C23 |
| hypercenter | C22×Q8 | D3 |
| hypocenter | C3 | #391 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(9,11)(12,14,15,19)(13,16,17,18);(12,13,15,17)(14,18,19,16);(2,3)(4,5,6,7)(8,9)(10,11)Polycyclic presentation:
2^6,3|a^2=d,b^2=f,c^2=f,c^a=ce,g^a=g^2,c^b=cf48 conjugacy classes — view the full character table.
| maximal subgroup of | #67699, #70100, #70102, … |
| maximal normal subgroup of | #67699, #70100, #70102, … |
| maximal quotient of | #71563, #71564, #71565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 788);