C₂×C₃×#10624749 #72244202

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁸

Aliases

direct product

C₂×C₃×#10624749

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴ ◁ G

4948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₃×#13091120
maximal subgroup of	C₂×#72221882, C₂×#72221849, AGL(1,𝔽₇)×#10624749, …
maximal normal subgroup of	C₂×#72221882, C₂×#72221849, C₂×C₃×#13091120
maximal quotient of	AGL(1,𝔽₇)×#10624749

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

C2×C3×#10624749 #72244202