C₃×#16864281 #27166158

order	12876
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#16864281

Splittings

Fitting, socle, Hall	C₃×C₂₉×C₃₇ ⋊ C₄
derived, Hall	C₂₉×C₃₇ ⋊ C₃×C₄
Hall	C₃×C₃₇ ⋊ #1181
Hall	C₃×C₂₉ ⋊ #3866
Sylow	C₃₇ ⋊ C₃×#1181
Sylow	C₂₉ ⋊ C₃×#3866

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#16864281
commutator, hypocenter	C₂₉×C₃₇	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₂₉×C₃₇	C₄
prime radical	C₃×D₁₀₇₃	C₂
cosocle	D₁₀₇₃	C₂×C₃
LCS	G ▷ C₂₉×C₃₇
UCS	C₁ ◁ C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3219

A minimal faithful permutation representation — 2 generators on 69 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#50311129, C₉×#16864281, C₃×#16864281⋊C₃, …
maximal normal subgroup of	#50311129, C₉×#16864281, C₃×#16864281⋊C₃, …
maximal quotient of	C₉×#16864281, C₃²×#16864281, C₃×C₃⋊#16864281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 27);

C3×#16864281 #27166158