C₃⋊(C₇⋊C₁₆) #66536

order	336
isoclinism	isoclinic class of D₂₁
automorphism	?
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₃⋊(C₇⋊C₁₆)
unambiguous semidirect product	C₇⋊(C₃⋊C₁₆)

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₂₁
commutator, hypocenter	C₃×C₇	C₁₆
Fitting, cosocle	C₃×C₇×C₈	C₂
socle, prime radical	C₂×C₃×C₇	C₈
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₈

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

(2,3)(4,5)(6,7)(9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(1,2,4,6,7,5,3)(8,9,10)

Polycyclic presentation:

2^4,3,7|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=f^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 5);