Finite Group DatabasebetaFinite Group Databasebeta| order | 244 |
| isoclinism | isoclinic class of #6923 |
| automorphism | Aut(G) = AGL(1,𝔽61) (order 3660) · Inn(G) = G · Out(G) = C3×C5 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C61 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C61 | C4 |
| prime radical, cosocle | D61 | C2 |
| LCS | G ▷ C61 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 61 |
A minimal faithful permutation representation — 2 generators on 61 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,49)(36,51,52,60)(40,55,56,58)(44,59,57,54)(46,61,53,50);(1,2,5,10,19,32,49,61,42,25,12,3,7,14,23,36,53,45,60,59,38,21,8,16,27,40,57,43,28,41,58,55,34,17,29,44,56,39,26,15,24,37,54,51,30,46,52,35,22,13,6,11,20,33,50,47,48,31,18,9,4)Polycyclic presentation:
2^2,61|a^2=b,c^a=c^11,c^b=c^6019 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6923, #6923⋊C3, C3×#6923, … |
| maximal normal subgroup of | C2×#6923, #6923⋊C3, C3×#6923, … |
| maximal quotient of | #92420, C2×#6923, C3×#6923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(244, 3);