C₃×#72215280 #72242424

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

direct product

C₃×#72215280

Splittings

Fitting	C₂×C₃×H(5,𝔽₂) ⋊ D₃²
derived	Ω⁺(4,𝔽₃) ⋊ C₂³×C₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂×S₄²
commutator, cosocle, hypocenter	Ω⁺(4,𝔽₃)	C₂³×C₃
Frattini	C₂	C₂×C₃×S₄²
Fitting	C₂×C₃×H(5,𝔽₂)	D₃²
hypercenter	C₂²×C₃	S₄²
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

3121 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3121 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 51
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 51 points:

(1,41,24)(2,8)(3,20,43)(4,39,18)(5,46,21)(6,42,16)(7,15,44)(9,33,31,27,28,32)(10,11,13,19,25,12)(14,45,36,34,47,48)(17,23,40)(22,38,26)(29,30)(35,37)(49,50,51);(3,15)(4,20)(5,22)(6,23)(9,14)(10,19)(11,12)(13,25)(16,42)(17,40)(18,41)(21,46)(24,43)(26,38)(27,34)(28,37)(29,45)(30,47)(31,36)(32,48)(33,35)(39,44);(1,2)(3,14)(4,19)(5,12)(6,25)(7,8)(9,15)(10,20)(11,22)(13,23)(16,37)(17,45)(18,34)(21,47)(24,48)(26,35)(27,41)(28,42)(29,40)(30,46)(31,39)(32,43)(33,38)(36,44)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((1,41,24)(2,8)(3,20,43)(4,39,18)(5,46,21)(6,42,16)(7,15,44)(9,33,31,27,28,32)(10,11,13,19,25,12)(14,45,36,34,47,48)(17,23,40)(22,38,26)(29,30)(35,37)(49,50,51), (3,15)(4,20)(5,22)(6,23)(9,14)(10,19)(11,12)(13,25)(16,42)(17,40)(18,41)(21,46)(24,43)(26,38)(27,34)(28,37)(29,45)(30,47)(31,36)(32,48)(33,35)(39,44), (1,2)(3,14)(4,19)(5,12)(6,25)(7,8)(9,15)(10,20)(11,22)(13,23)(16,37)(17,45)(18,34)(21,47)(24,48)(26,35)(27,41)(28,42)(29,40)(30,46)(31,39)(32,43)(33,38)(36,44));