Finite Group DatabasebetaFinite Group Databasebeta| order | 100 |
| isoclinism | isoclinic class of C5⋊D5 |
| automorphism | Aut(G) = C2×AGL(2,𝔽5) (order 24000) · Inn(G) = C5⋊D5 · Out(G) = C2×A5⋊C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| derived, Sylow | C52 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊D5 |
| commutator, hypocenter | C52 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C52 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3,4)(6,7)(8,9)(11,12)(13,14);(10,11,13,14,12);(5,6,8,9,7)Polycyclic presentation:
2^2,5^2|a^2=b,c^a=c^4,d^a=d^428 conjugacy classes — view the full character table.
| derived subgroup of | #87837, #87838, #87839, … |
| maximal subgroup of | #6033, #6034, D5×Dic5, … |
| maximal normal subgroup of | #6033, #6034, D5×Dic5, … |
| maximal quotient of | #6029, C2×#1039, C3×#1039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(100, 7);