C₅×#3480⋊C₃ #14511340

order	1920
isoclinism	isoclinic class of #548⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product	C₅×#3480⋊C₃
unambiguous semidirect product	#3480⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3480)⋊C₃

Splittings

Fitting, Hall	C₅×#3480 ⋊ C₃
Sylow	#3480 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃×C₅
Frattini	C₂²	C₂³×C₅×A₄
Fitting	C₅×#3480	C₃
socle	C₂²×C₅	C₂³×A₄
hypercenter	C₅×#117	A₄
hypocenter	Q₈	C₃×C₅×#41
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#117

564 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237038);