C₂²×C₁₉⋊D₁₉ #72227336

order	2888
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₁₉

Aliases

direct product

C₂²×C₁₉⋊D₁₉

Splittings

Fitting, socle	C₂²×C₁₉² ⋊ C₂
derived, Sylow	C₁₉² ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	C₁₉⋊D₁₉
commutator, cosocle, hypocenter	C₁₉²	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉²	C₂
LCS	G ▷ C₁₉²
UCS	C₁ ◁ C₂²

352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 1444

A minimal faithful permutation representation — 3 generators on 42 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#14524206, #72227354
maximal normal subgroup of	C₂²×#14524206, #72227354

GAP

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

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

C22×C19⋊D19 #72227336