Finite Group DatabasebetaFinite Group Databasebeta| order | 424 |
| isoclinism | isoclinic class of #6157 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C53 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #6157 |
| commutator, hypocenter | C53 | C8 |
| Fitting, socle, prime radical | C2×C53 | C4 |
| cosocle | Dic53 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 424 |
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)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53)(54,55,56,58,57,59,60,61);(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:
2^3,53|a^2=b,b^2=c,d^a=d^30,d^b=d^5234 conjugacy classes — view the full character table.
| maximal subgroup of | D53⋊C8, #11719936, C2×#88324, … |
| maximal normal subgroup of | D53⋊C8, #11719936, C2×#88324, … |
| maximal quotient of | #11719910, C2×#88324, C3×#88324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(424, 3);