C₃×#135758 #72225814

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₃×#135758

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1844
commutator	C₄×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
prime radical	C₂³×C₃	#492
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Conjugacy classes & character table

276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×#135758, C₃×#72213983, #72225880, …
maximal normal subgroup of	D₃×#135758, C₃×#72213983, #72225880, …
maximal quotient of	C₃×#72214006

GAP

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

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

C3×#135758 #72225814