Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of #3874 |
| automorphism | Aut(G) = C2×#64480⋊C2 (order 1200) · Inn(G) = #3874 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Hall | C52⋊C3 ⋊ C4 |
| Sylow | C52 ⋊ Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #3874 |
| commutator, hypocenter | C52⋊C3 | C4 |
| Fitting, socle | C2×C52 | D3 |
| prime radical, cosocle | C2×C52⋊C3 | C2 |
| LCS | G ▷ C52⋊C3 | |
| UCS | C1 ◁ C2 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,7)(2,4,6,13,5,9,12,10,11,14)(3,15)(16,17,18,19);(1,2,4)(3,6,14)(5,10,8)(7,11,13)(9,15,12)Polycyclic presentation:
2^2,3,5^2|a^2=b,c^a=c^2,e^a=d^4e^4,d^c=de^3,e^c=d^4e^326 conjugacy classes — view the full character table.
| maximal subgroup of | #10597076, #10597077, #10597079, … |
| maximal normal subgroup of | #10597076, #10597077, #10597079, … |
| maximal quotient of | #10597064, C2×#64470, #11744607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 13);