C₃×C₅×Dic₆₇ #16860831

order	4020
isoclinism	isoclinic class of D₆₇
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×Dic₆₇

Splittings

derived, Sylow	C₆₇ ⋊ C₃×C₄×C₅
Hall	C₃×C₅×C₆₇ ⋊ C₄
Hall	C₅×C₆₇ ⋊ C₃×C₄
Hall	C₃×C₆₇ ⋊ C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₆₇
commutator, hypocenter	C₆₇	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×D₆₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₆₇	C₂
cosocle	C₂×C₆₇	C₂×C₃×C₅
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₃×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 4020

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);(9,11,10,12)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Conjugacy classes & character table

1050 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×C₉×Dic₆₇, C₅×Dic₆₇⋊C₉, C₃×C₅×(C₆₇⋊C₃)⋊C₄, …
maximal normal subgroup of	C₅×C₉×Dic₆₇, C₅×Dic₆₇⋊C₉, C₃×C₅×(C₆₇⋊C₃)⋊C₄, …
maximal quotient of	C₅×C₉×Dic₆₇, C₃²×C₅×Dic₆₇, C₃×C₅×Dic₂₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4020, 3);

C3×C5×Dic67 #16860831