C₂⁵×D₃ #5983

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 1919877120
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product	C₂⁵×D₃
unambiguous semidirect product	C₃⋊C₂⁶
unambiguous semidirect product	(C₂×C₃)⋊C₂⁵

Fitting, socle	C₂⁵×C₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂⁶

	subgroup	G/H
center, hypercenter	C₂⁵	D₃
commutator, cosocle, hypocenter	C₃	C₂⁶
Frattini	C₁	G
Fitting, socle	C₂⁵×C₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁵

5650 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 6 generators on 13 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(12,13);(9,10)(11,12,13);(7,8);(5,6);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1542);