C₃⋊#1057 #64598

order	312
isoclinism	isoclinic class of C₃⋊#261
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = C₃⋊#261 · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊#261
commutator, hypocenter	C₃×C₁₃	C₈
Fitting, socle, prime radical	C₂×C₃×C₁₃	C₄
cosocle	C₃×Dic₁₃	C₂
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₂

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^3,3,13|a^2=b,b^2=c,d^a=d^2,e^a=e^5,e^b=e^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 14);