C₃×C₇×D₈₉ #16836644

order	3738
isoclinism	isoclinic class of D₈₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₇×D₈₉
unambiguous semidirect product	C₈₉⋊(C₂×C₃×C₇)

Splittings

Fitting, socle, Hall	C₃×C₇×C₈₉ ⋊ C₂
derived, Sylow	C₈₉ ⋊ C₂×C₃×C₇
Hall	C₇×C₈₉ ⋊ C₂×C₃
Hall	C₃×C₈₉ ⋊ C₂×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	D₈₉
commutator, cosocle, hypocenter	C₈₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₈₉	C₂
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₃×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 1869

A minimal faithful permutation representation — 2 generators on 99 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);(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×C₉×D₈₉, C₇⋊C₉×D₈₉, C₃×ΓL(1,𝔽₈)×D₈₉, …
maximal normal subgroup of	C₇×C₉×D₈₉, C₇⋊C₉×D₈₉, C₃×ΓL(1,𝔽₈)×D₈₉, …
maximal quotient of	C₇×C₉×D₈₉, C₃×C₇×D₂₆₇, C₃²×C₇×D₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3738, 3);

C3×C7×D89 #16836644