C₃×#144415 #72255259

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#144415

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₃×#3484	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

1884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,5,6)(3,7)(4,8)(9,10)(11,19,20,15)(12,16,21,24)(13,23)(14,18)(17,22)(25,26,27);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,13)(14,20)(16,23)(17,24)(18,19)(21,22);(1,4)(2,3)(5,7)(6,8)(9,12,22,11)(10,16,23,19)(13,20,14,21)(15,18,24,17)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#144415, #72270536
maximal subgroup of	D₃×#144415, C₃×#72213293, #72270536
maximal normal subgroup of	D₃×#144415, C₃×#72213293, #72270536

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,5,6)(3,7)(4,8)(9,10)(11,19,20,15)(12,16,21,24)(13,23)(14,18)(17,22)(25,26,27), (1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,13)(14,20)(16,23)(17,24)(18,19)(21,22), (1,4)(2,3)(5,7)(6,8)(9,12,22,11)(10,16,23,19)(13,20,14,21)(15,18,24,17));

C3×#144415 #72255259