Finite Group DatabasebetaFinite Group Databasebeta| order | 375 |
| isoclinism | isoclinic class of H(3,𝔽5)⋊C3 |
| automorphism | Aut(G) = AΓL(1,𝔽25) (order 1200) · Inn(G) = C52⋊C3 · Out(G) = M16 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | H(3,𝔽5)⋊C3 |
| Fitting, derived, Sylow | H(3,𝔽5) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C5 | C52⋊C3 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽5) | C3 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 75 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,5)(3,9,25)(4,8,16)(6,17,44)(7,14,35)(10,28,45)(11,32,63)(12,18,47)(13,24,38)(15,40,64)(19,49,59)(20,46,60)(21,52,72)(22,56,30)(23,39,34)(26,43,68)(27,37,31)(29,58,71)(33,53,57)(36,50,41)(42,65,74)(48,67,62)(51,73,61)(54,69,66)(55,70,75);(1,11,10,61,40)(2,7,53,6,64)(3,31,65,68,4)(5,73,25,52,37)(8,23,28,20,26)(9,48,74,21,50)(12,62,29,59,13)(14,15,56,38,71)(16,42,47,70,67)(17,39,43,57,41)(18,22,58,55,54)(19,24,32,27,51)(30,44,72,36,35)(33,60,75,66,34)(45,69,46,63,49)Polycyclic presentation:
3,5^3|b^a=bc^3,c^a=b^4c^3,c^b=cd23 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(375, 2);