Dic₃⋊(C₃⋊C₈) #63571

order	288
isoclinism	isoclinic class of #625
automorphism	Aut(G) = C₂⁵×D₃² (order 1152) · Inn(G) = D₃² · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₃² ⋊ C₄⋊C₈

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃²
commutator	C₂×C₃²	C₂×C₈
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	C₂×C₄
hypocenter	C₃²	C₄⋊C₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄

87 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^5,3^2|a^2=d,b^2=c,c^2=e,b^a=bd,g^a=g^2,f^b=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 219);