C₃×C₅×D₁₀₇ #15692011

order	3210
isoclinism	isoclinic class of D₁₀₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×D₁₀₇
unambiguous semidirect product	C₁₀₇⋊(C₂×C₃×C₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₀₇ ⋊ C₂
derived, Sylow	C₁₀₇ ⋊ C₂×C₃×C₅
Hall	C₅×C₁₀₇ ⋊ C₂×C₃
Hall	C₃×C₁₀₇ ⋊ C₂×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₁₀₇
commutator, cosocle, hypocenter	C₁₀₇	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₀₇	C₂
LCS	G ▷ C₁₀₇
UCS	C₁ ◁ C₃×C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 1605

A minimal faithful permutation representation — 2 generators on 115 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₂₅×D₁₀₇, C₃×C₅×D₅₃₅, C₃×C₅²×D₁₀₇, …
maximal normal subgroup of	C₃×C₂₅×D₁₀₇, C₃×C₅²×D₁₀₇, C₃×C₅×C₇×D₁₀₇, …
maximal quotient of	C₃×C₂₅×D₁₀₇, C₃×C₅×D₅₃₅, C₃×C₅²×D₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3210, 2);

C3×C5×D107 #15692011