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 ⋊ #543 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C2×C3×C22⋊C4 | C2 |
| prime radical | C24×C3 | C22 |
| hypercenter | C2×C22⋊C4 | D3 |
| hypocenter | C3 | #543 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(6,10)(8,11)(12,14,15,19)(13,16,17,18);(5,9)(8,11);(4,6)(5,8)(7,10)(9,11);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^6,3|a^2=e,d^2=e,c^a=cf,d^a=de,g^a=g^2,d^b=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #68114, #68115, #68116, … |
| maximal normal subgroup of | #68114, #68115, #68116, … |
| maximal quotient of | #73334, #73357, #73359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1042);