C₃×#16864280 #27166157

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

Aliases

direct product

C₃×#16864280

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	#16864280
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)(10,26,18,17)(14,22,19,21)(16,20,24,23)(25,29,31,30)(27,28,33,32)(34,35,37,36)(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);(1,2,5,10,19,13,6,11,20,28,31,24,15,22,29,35,37,33,26,17,27,34,36,30,21,8,16,25,32,23,12,3,7,14,18,9,4)(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	#50311128, C₉×#16864280, C₃×#16864280⋊C₃, …
maximal normal subgroup of	#50311128, C₉×#16864280, C₃×#16864280⋊C₃, …
maximal quotient of	C₉×#16864280, C₃²×#16864280, C₃×C₃⋊#16864280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 26);

C3×#16864280 #27166157