C₃×D₅×C₇₁ #14523804

order	2130
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅×C₇₁
unambiguous semidirect product	C₅⋊(C₂×C₃×C₇₁)

Splittings

Fitting, socle, Hall	C₃×C₅×C₇₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃×C₇₁
Hall	C₅×C₇₁ ⋊ C₂×C₃
Hall	C₃×C₅ ⋊ C₂×C₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₇₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₇₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1065

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(5,8)(6,7)

Conjugacy classes & character table

852 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₅)×C₇₁, C₃×C₇₁⋊AGL(1,𝔽₅), C₃×D₅×D₇₁, …
maximal normal subgroup of	C₃×AGL(1,𝔽₅)×C₇₁, C₃×C₇₁⋊AGL(1,𝔽₅), C₃×D₅×D₇₁, …
maximal quotient of	C₃×Dic₅×C₇₁, C₂×C₃×D₅×C₇₁, C₃×C₇×D₅×C₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2130, 4);

C3×D5×C71 #14523804