C₃×#107130 #72270331

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#107130

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#379
commutator	C₂²×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³×C₃
prime radical	C₂⁴×C₃×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

5764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270414, #72270420, D₃×#107130, …
maximal subgroup of	C₃×#72219163, #72270414, #72270420, …
maximal normal subgroup of	C₃×#72219163, #72270414, #72270420, …

GAP

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

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

C3×#107130 #72270331