C₃⋊#1055 #64589

order	312
isoclinism	isoclinic class of D₃₉
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃₉
commutator, hypocenter	C₃×C₁₃	C₈
Fitting, cosocle	C₃×C₄×C₁₃	C₂
socle, prime radical	C₂×C₃×C₁₃	C₄
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₄

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 5);