C₂³×D₁₃ #6139

order	208
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) order 209664
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×D₁₃
unambiguous semidirect product	C₁₃⋊C₂⁴
unambiguous semidirect product	(C₂×C₁₃)⋊C₂³

Fitting, socle	C₂³×C₁₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂³

134 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2)(3,4)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19);(5,6)(7,8,10,12,14,16,18,19,17,15,13,11,9);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 50);