C₃×C₇×D₇₉ #15693289

order	3318
isoclinism	isoclinic class of D₇₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₇×D₇₉

Splittings

Fitting, socle, Hall	C₃×C₇×C₇₉ ⋊ C₂
derived, Sylow	C₇₉ ⋊ C₂×C₃×C₇
Hall	C₇×C₇₉ ⋊ C₂×C₃
Hall	C₃×C₇₉ ⋊ C₂×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	D₇₉
commutator, cosocle, hypocenter	C₇₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₉	C₂
LCS	G ▷ C₇₉
UCS	C₁ ◁ C₃×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 1659

A minimal faithful permutation representation — 2 generators on 89 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);(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₅×C₇×D₇₉, C₃×C₇×D₃₉₅, C₃×C₄₉×D₇₉, …
maximal normal subgroup of	C₃×C₅×C₇×D₇₉, C₃×C₄₉×D₇₉, C₃×C₇²×D₇₉, …
maximal quotient of	C₃×C₅×C₇×D₇₉, C₃×C₇×D₃₉₅, C₃×C₄₉×D₇₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 6);

C3×C7×D79 #15693289