C₃×#498888 #72254284

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

Aliases

direct product

C₃×#498888

Subgroups

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

4668 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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#498888, #72269931
maximal subgroup of	C₃×#72215601, D₃×#498888, #72269931
maximal normal subgroup of	C₃×#72215601, D₃×#498888, #72269931

GAP

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

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

C3×#498888 #72254284