Dic₃₀ #1216

order	120
isoclinism	isoclinic class of Dic₃₀
automorphism	Aut(G) = D₃×D₄×AGL(1,𝔽₅) (order 960) · Inn(G) = C₂×D₁₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₃₀

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅
commutator, socle, prime radical, cosocle	C₂×C₃×C₅	C₂²
Fitting	C₃×C₄×C₅	C₂
hypercenter	C₄	D₁₅
hypocenter	C₃×C₅	Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^3,3,5|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 26);