Finite Group Databasebeta

← Prev (#72228922) Next (#72228924) →

3₋¹⁺²×D₁₃⋊C₃ #72228923

order	2106
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃²

Aliases

3₋¹⁺²×D₁₃⋊C₃

Splittings

Fitting	C₁₃×3₋¹⁺² ⋊ C₂×C₃
Hall	3₋¹⁺²×C₁₃⋊C₃ ⋊ C₂
Hall	D₁₃ ⋊ C₃×3₋¹⁺²
Sylow	C₁₃ ⋊ C₂×C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃²×D₁₃⋊C₃
commutator, socle, cosocle	C₃×C₁₃	C₂×C₃³
Fitting	C₁₃×3₋¹⁺²	C₂×C₃
prime radical	C₃²×D₁₃⋊C₃	C₃
hypercenter	3₋¹⁺²	D₁₃⋊C₃
hypocenter	C₁₃	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,11,13,12,4)(3,7,6,5,9,10);(15,18,19)(17,21,22);(2,4,5)(3,7,8)(6,10,11)(9,12,13)(14,19,22,16,18,17,20,15,21)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228932, 3₋¹⁺²×AGL(1,𝔽₁₃), AGL(1,ℤ/9)×D₁₃⋊C₃, …
maximal normal subgroup of	3₋¹⁺²×AGL(1,𝔽₁₃), AGL(1,ℤ/9)×D₁₃⋊C₃, #72228957, …
maximal quotient of	#72228932, C₂×3₋¹⁺²×D₁₃⋊C₃

GAP

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

gap> G := Group((1,2,11,13,12,4)(3,7,6,5,9,10), (15,18,19)(17,21,22), (2,4,5)(3,7,8)(6,10,11)(9,12,13)(14,19,22,16,18,17,20,15,21));