C₁₃×C₄²⋊C₃ #10597622

order	624
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₄

Aliases

direct product	C₁₃×C₄²⋊C₃
unambiguous semidirect product	C₄²⋊(C₃×C₁₃)

Fitting, Hall	C₄²×C₁₃ ⋊ C₃
derived, Sylow	C₄² ⋊ C₃×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	C₄²⋊C₃
commutator, cosocle, hypocenter	C₄²	C₃×C₁₃
Frattini	C₂²	A₄×C₁₃
Fitting	C₄²×C₁₃	C₃
socle	C₂²×C₁₃	A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₁₃

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 59);