C₂²⋊GL(2,𝔽₅) #14514930

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

non-solvable

Aliases

unambiguous semidirect product

C₂²⋊GL(2,𝔽₅)

Subgroups

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

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

normal subgroups only

This group has 433 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×GL(2,𝔽₅), #72275838
maximal normal subgroup of	D₄×GL(2,𝔽₅)
maximal quotient of	#72276774, #72276971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 240628);