C₃×#498885 #72248158

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#498885

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
socle	C₂²×C₃	C₂×B(4,𝔽₂)
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

4652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#498885, #72254161, #72270962
maximal subgroup of	D₃×#498885, #72254161, C₃×#72215601, …
maximal normal subgroup of	D₃×#498885, #72254161, C₃×#72215601, …

GAP

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

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

C3×#498885 #72248158