C₃⋊(C₇⋊C₈) #4235

order	168
isoclinism	isoclinic class of D₂₁
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₂₁ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₃⋊(C₇⋊C₈)
unambiguous semidirect product	C₇⋊(C₃⋊C₈)

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₂₁
commutator, hypocenter	C₃×C₇	C₈
Fitting, cosocle	C₃×C₄×C₇	C₂
socle, prime radical	C₂×C₃×C₇	C₄
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^3,3,7|a^2=b,b^2=c,d^a=d^2,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 5);