Finite Group DatabasebetaFinite Group Databasebeta| order | 100 |
| isoclinism | isoclinic class of #1042 |
| automorphism | Aut(G) = AGL(1,𝔽5)2 (order 400) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C52 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52 | C4 |
| prime radical, cosocle | C5×D5 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 | |
13 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 10 points:
(2,3)(4,5)(7,8,9,10);(1,2,4,5,3)(6,7,10,8,9)Polycyclic presentation:
2^2,5^2|a^2=b,c^a=c^4,d^a=d^3,d^b=d^413 conjugacy classes — view the full character table.
| maximal subgroup of | D5×AGL(1,𝔽5), C2×#1042, C3×#1042, … |
| maximal normal subgroup of | D5×AGL(1,𝔽5), C2×#1042, C3×#1042, … |
| maximal quotient of | #6032, C2×#1042, C3×#1042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(100, 10);