C₃×C₅×D₁₂₁ #16813786

order	3630
isoclinism	isoclinic class of D₁₂₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×D₁₂₁

Splittings

Fitting, Hall	C₃×C₅×C₁₂₁ ⋊ C₂
derived, Sylow	C₁₂₁ ⋊ C₂×C₃×C₅
Hall	C₅×C₁₂₁ ⋊ C₂×C₃
Hall	C₃×C₁₂₁ ⋊ C₂×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₁₂₁
commutator, cosocle, hypocenter	C₁₂₁	C₂×C₃×C₅
Frattini	C₁₁	C₃×C₅×D₁₁
Fitting	C₃×C₅×C₁₂₁	C₂
socle	C₃×C₅×C₁₁	D₁₁
LCS	G ▷ C₁₂₁
UCS	C₁ ◁ C₃×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 1815

A minimal faithful permutation representation — 2 generators on 129 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₉×D₁₂₁, C₃²×C₅×D₁₂₁, C₃×C₅×D₃₆₃, …
maximal normal subgroup of	C₅×C₉×D₁₂₁, C₃²×C₅×D₁₂₁, C₃×D₁₂₁⋊C₂₅, …
maximal quotient of	C₅×C₉×D₁₂₁, C₃²×C₅×D₁₂₁, C₃×C₅×D₃₆₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3630, 6);

C3×C5×D121 #16813786