C₃×#94425 #72271735

order	1536
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#94425

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#7304
commutator	C₂²×D₄	C₃×C₄²
Frattini, cosocle	C₂×#535	C₂²×C₃
prime radical	C₂×C₃×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ C₂×C₃×#535 ◁ G

942 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3)(4,5,8,14,9,16,7,22)(6,17,20,11)(10,15,25,12,26,13,24,30)(18,28,34,32)(19,33,35,29)(21,31,23,27);(4,6,7,23)(5,11,22,32)(8,18,9,19)(10,21,24,35)(12,17,13,27)(14,31,16,29)(15,28,30,33)(20,26,34,25)

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72216507
maximal normal subgroup of	C₃×#72216507

GAP

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

gap> G := Group((1,2,3)(4,5,8,14,9,16,7,22)(6,17,20,11)(10,15,25,12,26,13,24,30)(18,28,34,32)(19,33,35,29)(21,31,23,27), (4,6,7,23)(5,11,22,32)(8,18,9,19)(10,21,24,35)(12,17,13,27)(14,31,16,29)(15,28,30,33)(20,26,34,25));