ΓL(1,𝔽₈)⋊C₁₆ #66532

order	336
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = C₂×C₄×AGL(1,𝔽₇) (order 336) · Inn(G) = AGL(1,𝔽₇) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	ΓL(1,𝔽₈)⋊C₁₆
unambiguous semidirect product	(C₇⋊C₁₆)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₁₆
Fitting, cosocle	C₇×C₈	C₂×C₃
socle	C₂×C₇	C₃×C₈
prime radical	C₂×ΓL(1,𝔽₈)	C₈
LCS	G ▷ C₇
UCS	C₁ ◁ C₈

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 1);