C₂×C₃×#10624934 #72244182

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁵ · Schur multiplier (H₂) = C₃¹⁰

Aliases

direct product

C₂×C₃×#10624934

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃⁵
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×3₋¹⁺² ◁ G

12580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221906, C₂×#72221894, C₂×C₃×#13091662
maximal subgroup of	C₂×#72221882, C₂×#72221906, C₂×#72221894, …
maximal normal subgroup of	C₂×#72221882, C₂×#72221906, C₂×#72221894, …

GAP

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

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

C2×C3×#10624934 #72244182