C₅×#67997 #14510656

order	1920
isoclinism	isoclinic class of #4621
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

solvable

Aliases

direct product

C₅×#67997

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	#857
commutator, hypocenter	#95⋊C₃	C₄×C₅
Frattini	C₂²×C₄	C₅×S₄
Fitting	C₅×#376	D₃
socle	C₂³×C₅	SL(2,ℤ/4)
prime radical	C₅×#95⋊C₃	C₄
cosocle	#376⋊C₃	C₂×C₅
LCS	G ▷ #95⋊C₃
UCS	C₁ ◁ C₄×C₅

140 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236354);