C₂³×D₁₉ #64552

order	304
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×D₁₉
unambiguous semidirect product	C₁₉⋊C₂⁴
unambiguous semidirect product	(C₂×C₁₉)⋊C₂³

Fitting, socle	C₂³×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂³

134 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 152

A minimal faithful permutation representation — 4 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 41);