C₃×Dic₂₅ #64459

order	300
isoclinism	isoclinic class of D₂₅
automorphism	Aut(G) = C₂²×AGL(1,ℤ/25) (order 2000) · Inn(G) = D₂₅ · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₂₅

Splittings

derived, Sylow	C₂₅ ⋊ C₃×C₄
Hall	C₃×C₂₅ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₂₅
commutator, hypocenter	C₂₅	C₃×C₄
Frattini	C₂×C₅	C₃×D₅
Fitting	C₂×C₃×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅	D₅
cosocle	C₂×C₂₅	C₂×C₃
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₂×C₃

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 300

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(4,6,5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#6016, C₃⋊#6016, #10597026, …
maximal normal subgroup of	C₃×#6016, C₃⋊#6016, #10597026, …
maximal quotient of	C₃×#6014, C₂×C₃×Dic₂₅, C₉×Dic₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 2);

C3×Dic25 #64459

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×Dic₂₅ #64459