C₃×C₅×D₇₁ #14523803

order	2130
isoclinism	isoclinic class of D₇₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×D₇₁

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₇₁
commutator, cosocle, hypocenter	C₇₁	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇₁	C₂
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₃×C₅

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×D₅×D₇₁, C₅×D₃×D₇₁, D₁₅×D₇₁, …
maximal normal subgroup of	C₃×D₅×D₇₁, C₅×D₃×D₇₁, D₁₅×D₇₁, …
maximal quotient of	C₃×C₅×Dic₇₁, C₂×C₃×C₅×D₇₁, C₃×C₅×C₇×D₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2130, 3);

C3×C5×D71 #14523803