Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of #64480 |
| automorphism | Aut(G) = AΓL(1,𝔽25) (order 1200) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, Sylow | C52 ⋊ Dic3 |
| derived, Hall | C52⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C52⋊C3 | C4 |
| Fitting, socle | C52 | Dic3 |
| prime radical, cosocle | (C5⋊D5)⋊C3 | C2 |
| LCS | G ▷ C52⋊C3 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 15 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,3,8,7)(2,11,5,14)(4,15)(6,10,12,13);(1,2,4)(3,5,11)(6,14,8)(7,15,10)(9,12,13)Polycyclic presentation:
2^2,3,5^2|a^2=b,c^a=c^2,d^a=d^3e^4,e^a=e^2,d^b=d^4,e^b=e^4,d^c=d^3e^2,e^c=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 12 | 12 | 25 | 50 | 50 | 75 | 75 |
| order | 1 | 5 | 5 | 2 | 3 | 6 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | -1 | 1 | -1 | ||
| χ4 | 1 | 1 | 1 | -1 | 1 | -1 | ||
| χ5 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 |
| χ6 | 2 | 2 | 2 | -2 | -1 | 1 | 0 | 0 |
| χ7 | 12 | 2 | -3 | 0 | 0 | 0 | 0 | 0 |
| χ8 | 12 | -3 | 2 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | #64480⋊C2, C2×#64480, C3×#64480, … |
| maximal normal subgroup of | #64480⋊C2, C2×#64480, C3×#64480, … |
| maximal quotient of | #10597074, C2×#64480, #11744634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 23);