C₃²×Dic₈₉ #15691963

order	3204
isoclinism	isoclinic class of D₈₉
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×Dic₈₉

Splittings

derived, Sylow	C₈₉ ⋊ C₃²×C₄
Hall	C₃²×C₈₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₈₉
commutator, hypocenter	C₈₉	C₃²×C₄
Frattini	C₂	C₃²×D₈₉
Fitting, socle, prime radical	C₂×C₃²×C₈₉	C₂
cosocle	C₂×C₈₉	C₂×C₃²
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₂×C₃²

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 3204

A minimal faithful permutation representation — 2 generators on 99 points:

(1,2,3)(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);(4,5,6)(7,9,8,10)(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

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

Relationships

maximal subgroup of	C₃²×C₅×Dic₈₉, C₃²×Dic₄₄₅, C₃²×#13184069, …
maximal normal subgroup of	C₃²×C₅×Dic₈₉, C₃²×C₇×Dic₈₉, C₃²×Dic₈₉⋊C₁₁, …
maximal quotient of	C₃²×C₅×Dic₈₉, C₃²×Dic₄₄₅, C₃²×#13184069, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3204, 14);

C32×Dic89 #15691963