C₅×#73129 #14510853

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73129

Subgroups

	subgroup	G/H
center	C₄×C₅	GL(2,ℤ/4)
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×#568	D₃
socle	C₂×C₅	SL(2,ℤ/4)⋊C₄
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₄⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

244 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5)(6,42,46,19,22,52,10,39,40,8,11,51,13,36,25,23,24,37,9,20,43,21,28,44)(7,35,45,18,32,49,15,29,41,14,34,53,17,47,26,31,27,50,12,48,33,30,16,38);(6,7,10,15,13,17,9,12)(8,18,19,30,21,31,23,14)(11,25,24,43,28,46,22,40)(16,33,32,45,34,41,27,26)(20,37,36,51,39,52,42,44)(29,38,35,50,48,53,47,49)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236551);