A₄×C₃⋊(D₁₉⋊C₃) #72276901

order	4104
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

A₄×C₃⋊(D₁₉⋊C₃)

Splittings

Fitting, socle, derived	C₂²×C₃×C₁₉ ⋊ C₂×C₃²
Sylow	C₁₉ ⋊ C₃×D₃×A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₃×C₁₉	C₂×C₃²
LCS	G ▷ C₂²×C₃×C₁₉
UCS	C₁

162 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×C₃⋊AGL(1,𝔽₁₉), D₃×A₄×D₁₉⋊C₃
maximal normal subgroup of	A₄×C₃⋊AGL(1,𝔽₁₉), D₃×A₄×D₁₉⋊C₃

GAP

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

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