C₃×#72214006 #72226186

order	3072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₃×#72214006

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#8813
commutator	C₈²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₈²	C₂³×C₃
prime radical	C₂³×C₃	#1705
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

846 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 394 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) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

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

GAP

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

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