C₃×D₂₀₆₇ #27137173

order	12402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₀₆₇
unambiguous semidirect product	C₃⋊(C₃×D₆₈₉)
unambiguous semidirect product	C₅₃⋊(C₃×D₃₉)

Splittings

Fitting, socle, Hall	C₃²×C₁₃×C₅₃ ⋊ C₂
derived	C₃×C₁₃×C₅₃ ⋊ C₂×C₃
Hall	C₁₃×C₅₃ ⋊ C₃×D₃
Hall	C₃²×C₅₃ ⋊ D₁₃
Hall	C₃²×C₁₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ C₃×D₃₉
Sylow	C₁₃ ⋊ C₃×D₁₅₉
Sylow	C₃² ⋊ D₆₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₂₀₆₇
commutator, cosocle, hypocenter	C₃×C₁₃×C₅₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₅₃	C₂
LCS	G ▷ C₃×C₁₃×C₅₃
UCS	C₁ ◁ C₃

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 4134

A minimal faithful permutation representation — 2 generators on 72 points:

(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)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68,69)(71,72);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,57,59,61,63,65,66,64,62,60,58,56)(70,71,72)

Conjugacy classes & character table

3105 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₃₉×D₅₃, C₃×D₁₃×D₁₅₉, C₃×D₃×D₆₈₉, …
maximal normal subgroup of	C₃×D₃₉×D₅₃, C₃×D₁₃×D₁₅₉, C₃×D₃×D₆₈₉, …
maximal quotient of	C₃×Dic₂₀₆₇, C₂×C₃×D₂₀₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 36);

C3×D2067 #27137173