C₃×C₅×D₁₁₃ #16810067

order	3390
isoclinism	isoclinic class of D₁₁₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×D₁₁₃
unambiguous semidirect product	C₁₁₃⋊(C₂×C₃×C₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₁₃ ⋊ C₂
derived, Sylow	C₁₁₃ ⋊ C₂×C₃×C₅
Hall	C₅×C₁₁₃ ⋊ C₂×C₃
Hall	C₃×C₁₁₃ ⋊ C₂×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₁₁₃
commutator, cosocle, hypocenter	C₁₁₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₁₃	C₂
LCS	G ▷ C₁₁₃
UCS	C₁ ◁ C₃×C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 1695

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₂₅×D₁₁₃, C₃×C₅×D₅₆₅, C₃×C₅²×D₁₁₃, …
maximal normal subgroup of	C₃×C₂₅×D₁₁₃, C₃×C₅²×D₁₁₃, C₃×C₅×C₇×D₁₁₃, …
maximal quotient of	C₃×C₂₅×D₁₁₃, C₃×C₅×D₅₆₅, C₃×C₅²×D₁₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3390, 2);

C3×C5×D113 #16810067