Finite Group DatabasebetaFinite Group Databasebeta| order | 114 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = AGL(1,𝔽19) (order 342) · Inn(G) = C19⋊C3 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = C2×#72214746 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C2×C19⋊C3 |
| unambiguous semidirect product | (C2×C19)⋊C3 |
| Fitting, socle, Hall | C2×C19 ⋊ C3 |
| derived, Sylow | C19 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C19⋊C3 |
| commutator, cosocle, hypocenter | C19 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C19 | C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 38 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(4,5,7)(6,9,10)(8,12,11)(13,17,15)(14,18,16)(19,21,20);(3,4,6,10,15,20,17,5,8,13,18,7,11,16,9,14,19,21,12)18 conjugacy classes — view the full character table.
| maximal subgroup of | (C19⋊C3)⋊C4, C4×C19⋊C3, C2×D19⋊C3, … |
| maximal normal subgroup of | (C19⋊C3)⋊C4, C4×C19⋊C3, C2×D19⋊C3, … |
| maximal quotient of | C4×C19⋊C3, C22×C19⋊C3, C2×#4294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(114, 2);