C₂×C₃³×C₄⋊Q₈ #13179986

order	1728
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃³ · Schur multiplier (H₂) = C₂⁵×C₃³

Aliases

C₂×C₃³×C₄⋊Q₈

	subgroup	G/H
center, socle, prime radical	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

4060 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 1728

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46049);