C₅×C₃⋊D₃×D₁₀₉ #34087169

order	19620
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₅×C₃⋊D₃×D₁₀₉
unambiguous semidirect product	(C₃×D₁₀₉)⋊(C₅×D₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₀₉ ⋊ C₂²
derived, Hall	C₃²×C₁₀₉ ⋊ C₂²×C₅
Hall	C₅×C₁₀₉ ⋊ C₂×C₃⋊D₃
Sylow	C₁₀₉ ⋊ C₂×C₅×C₃⋊D₃
Hall	C₃²×C₅ ⋊ C₂×D₁₀₉
Sylow	C₃² ⋊ C₂×C₅×D₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊D₃×D₁₀₉
commutator, cosocle, hypocenter	C₃²×C₁₀₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₀₉	C₂²
LCS	G ▷ C₃²×C₁₀₉
UCS	C₁ ◁ C₅

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 4905

A minimal faithful permutation representation — 3 generators on 120 points:

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

Conjugacy classes & character table

1680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19620, 137);

C5×C3⋊D3×D109 #34087169