D₂₁⋊C₄ #4244

order	168
isoclinism	isoclinic class of D₃×D₇
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₃×D₇ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

D₂₁⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂×C₄
Fitting, socle, cosocle	C₂×C₃×C₇	C₂²
prime radical	C₂×D₂₁	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 14);