C₅×#3428⋊C₃ #14511317

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product	C₅×#3428⋊C₃
unambiguous semidirect product	#3428⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3428)⋊C₃

Splittings

Fitting, Hall	C₅×#3428 ⋊ C₃
Sylow	#3428 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×A₄
commutator, hypocenter	Q₈	C₂²×C₃×C₄×C₅
Frattini	C₂²	C₂³×C₅×A₄
Fitting	C₅×#3428	C₃
socle	C₂²×C₅	C₂³×A₄
prime radical	C₂²×C₅×#41⋊C₃	C₂
cosocle	C₂×Q₈	C₂³×C₃×C₅
hypercenter	C₅×#117	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#117

730 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237015);