C₅×#73132 #14510856

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73132

Splittings

Fitting

C₅×#568 ⋊ D₃

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	(C₂×S₄)⋊C₄
prime radical	C₅×#5923	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

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

normal subgroups only

This group has 314 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

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

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, 236554);