A₄×C₃₇ #89229

order	444
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₄×C₉×S₄ (order 864) · Inn(G) = A₄ · Out(G) = C₂×C₄×C₉
homology	Abelianization (H₁) = C₃×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

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

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

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

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 148

A minimal faithful permutation representation — 2 generators on 41 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);(1,4)(2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(444, 13);