C₃×#128059 #72271645

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#128059

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#342
commutator	C₂²×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³×C₃
prime radical	C₂³×C₃×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×#41 ◁ G

3052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72271766, #72271768
maximal subgroup of	C₃×#72213489, #72271766, #72271768
maximal normal subgroup of	C₃×#72213489, #72271766, #72271768

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

C3×#128059 #72271645