C₃²×Dic₅₉ #14523772

order	2124
isoclinism	isoclinic class of D₅₉
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×Dic₅₉
unambiguous semidirect product	C₅₉⋊(C₃²×C₄)

Splittings

derived, Sylow	C₅₉ ⋊ C₃²×C₄
Hall	C₃²×C₅₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₅₉
commutator, hypocenter	C₅₉	C₃²×C₄
Frattini	C₂	C₃²×D₅₉
Fitting, socle, prime radical	C₂×C₃²×C₅₉	C₂
cosocle	C₂×C₅₉	C₂×C₃²
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₂×C₃²

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 2124

A minimal faithful permutation representation — 2 generators on 69 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);(4,5,6)(7,9,8,10)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×ΓL(1,𝔽₈)⋊Dic₅₉, C₃×ΓL(1,𝔽₈)×Dic₅₉, C₃²×C₇×Dic₅₉, …
maximal normal subgroup of	C₃²×C₇×Dic₅₉, C₃²×C₁₁×Dic₅₉, C₃²×C₁₃×Dic₅₉, …
maximal quotient of	C₃×ΓL(1,𝔽₈)⋊Dic₅₉, C₃×ΓL(1,𝔽₈)×Dic₅₉, C₃²×C₇×Dic₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2124, 12);

C32×Dic59 #14523772