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 ⋊ #544 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C2×C3×C4⋊C4 | C2 |
| prime radical | C23×C3 | C23 |
| hypercenter | C2×C4⋊C4 | D3 |
| hypocenter | C3 | #544 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
206 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)(6,10)(8,11)(12,14,15,19)(13,16,17,18);(5,9)(8,11);(4,6,7,10)(5,8,9,11);(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,b^2=f,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 | #68138, #68139, #68140, … |
| maximal normal subgroup of | #68138, #68139, #68140, … |
| maximal quotient of | #73334, #73350, #73360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1059);