#1057⋊C₃ #64591

order	312
isoclinism	isoclinic class of AGL(1,𝔽₁₃)
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₃) (order 312) · Inn(G) = AGL(1,𝔽₁₃) · Out(G) = C₂
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃	C₃×C₈
Fitting, socle	C₂×C₁₃	C₃×C₄
prime radical	C₂×C₁₃⋊C₃	C₄
cosocle	Dic₁₃	C₂×C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2,3,2^2,13|a^2=c,c^2=d,e^a=e^8,e^b=e^3,e^c=e^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 7);