C₃×C₃₇×#1181 #27166156

order	12876
isoclinism	isoclinic class of #1181
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₃₇×#1181

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₃₇	#1181
commutator, hypocenter	C₂₉	C₃×C₄×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₂₉×C₃₇	C₄
prime radical	C₃×C₃₇×D₂₉	C₂
cosocle	D₂₉	C₂×C₃×C₃₇
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₃×C₃₇

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)(10,25,18,17)(14,23,22,21)(16,20,19,24)(26,27,29,28)(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);(1,2,5,10,19,15,23,17,26,28,24,12,3,7,14,22,13,6,11,20,27,29,25,21,8,16,18,9,4)

Conjugacy classes & character table

1221 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#1181×#66524, C₉×C₃₇×#1181, C₃×C₃₇⋊C₃×#1181, …
maximal normal subgroup of	#1181×#66524, C₉×C₃₇×#1181, C₃×C₃₇⋊C₃×#1181, …
maximal quotient of	C₉×C₃₇×#1181, C₃²×C₃₇×#1181, C₃×C₃₇×C₃⋊#1181

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 25);

C3×C37×#1181 #27166156