Finite Group DatabasebetaFinite Group Databasebeta| order | 212 |
| isoclinism | isoclinic class of #6157 |
| automorphism | Aut(G) = AGL(1,𝔽53) (order 2756) · Inn(G) = G · Out(G) = C13 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C53 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C4 |
| prime radical, cosocle | D53 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 53 |
A minimal faithful permutation representation — 2 generators on 53 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)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53);(1,2,5,10,19,32,21,8,16,27,39,48,51,44,30,42,28,40,49,45,35,22,13,6,11,20,33,41,25,12,3,7,14,23,36,46,34,17,29,43,50,53,52,47,38,26,15,24,37,31,18,9,4)Polycyclic presentation:
2^2,53|a^2=b,c^a=c^30,c^b=c^5217 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6157, C3×#6157, C3⋊#6157, … |
| maximal normal subgroup of | C2×#6157, C3×#6157, C5×#6157, … |
| maximal quotient of | #88324, C2×#6157, C3×#6157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(212, 3);