#1055⋊C₃ #64585

order	312
isoclinism	isoclinic class of D₁₃⋊C₃
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₃) (order 624) · Inn(G) = D₁₃⋊C₃ · Out(G) = C₂³
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₃⋊C₃
commutator, hypocenter	C₁₃	C₃×C₈
Fitting, cosocle	C₄×C₁₃	C₂×C₃
socle	C₂×C₁₃	C₃×C₄
prime radical	C₂×C₁₃⋊C₃	C₄
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₄

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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 1);