C₃×#500177 #72270043

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#500177

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
prime radical	C₃×#52004	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

2848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270741, D₃×#500177
maximal subgroup of	C₃×#72210575, #72270741, D₃×#500177
maximal normal subgroup of	C₃×#72210575, #72270741, D₃×#500177

GAP

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

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

C3×#500177 #72270043