C₂×A₄×C₁₇ #88005

order	408
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₁₆×S₄ (order 384) · Inn(G) = A₄ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₂×C₃×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×A₄×C₁₇
unambiguous semidirect product	C₂³⋊(C₃×C₁₇)
unambiguous semidirect product	(C₂³×C₁₇)⋊C₃

	subgroup	G/H
center, hypercenter	C₂×C₁₇	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₂³×C₁₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₁₇

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 102

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 42);