C₄⋊GL(2,𝔽₅) #14514960

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

unambiguous semidirect product

C₄⋊GL(2,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×S₅
commutator	C₂×SL(2,𝔽₅)	C₂×C₄
Fitting, hypercenter	C₄²	S₅
socle	C₂²	C₂×A₅⋊C₄
prime radical, cosocle	C₂×#6732	C₂²
hypocenter	SL(2,𝔽₅)	C₄⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₅) ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

298 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Permutation generators:

(1,28)(2,27)(3,48,26,36,17,39,8,38,15,46,24,31,4,47,25,35,18,40,7,37,16,45,23,32)(5,29,21,52,9,43,14,34,11,41,19,50,6,30,22,51,10,44,13,33,12,42,20,49);(1,28,2,27)(3,43,16,46,5,34,24,52,9,40,4,44,15,45,6,33,23,51,10,39)(7,49,25,36,13,29,17,42,20,31,8,50,26,35,14,30,18,41,19,32)(11,48,12,47)(21,37,22,38)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×GL(2,𝔽₅), Q₈×GL(2,𝔽₅), #72276774, …
maximal normal subgroup of	D₄×GL(2,𝔽₅), Q₈×GL(2,𝔽₅), #72276774, …
maximal quotient of	#72276979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 240658);