A₄×C₂₉×C₃₇ #27166175

order	12876
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₉×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₂₉×C₃₇
unambiguous semidirect product	C₂²⋊(C₃×C₂₉×C₃₇)
unambiguous semidirect product	(C₂²×C₂₉)⋊(C₃×C₃₇)

Splittings

Fitting, socle, Hall	C₂²×C₂₉×C₃₇ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₂₉×C₃₇
Hall	C₂²×C₃₇ ⋊ C₃×C₂₉
Hall	C₂²×C₂₉ ⋊ C₃×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂₉×C₃₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₂₉×C₃₇
Frattini	C₁	G
Fitting, socle	C₂²×C₂₉×C₃₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂₉×C₃₇

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 4292

A minimal faithful permutation representation — 2 generators on 70 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(1,4)(2,3)

Conjugacy classes & character table

4292 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	A₄×C₂₉×C₃₇⋊C₃, C₃×A₄×C₂₉×C₃₇
maximal normal subgroup of	A₄×C₂₉×C₃₇⋊C₃, C₃×A₄×C₂₉×C₃₇
maximal quotient of	C₂₉×C₂²⋊C₉×C₃₇, C₃×A₄×C₂₉×C₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 44);

A4×C29×C37 #27166175