C₅×#3522⋊C₃ #14511357

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

solvable

Aliases

direct product	C₅×#3522⋊C₃
unambiguous semidirect product	#3522⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3522)⋊C₃

Splittings

Fitting, Hall	C₅×#3522 ⋊ C₃
Sylow	#3522 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂³×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃×C₅
Frattini	C₂²	C₂³×C₅×A₄
Fitting	C₅×#3522	C₃
hypercenter	C₅×C₂²⋊Q₈	A₄
hypocenter	Q₈	C₃×C₅×#41
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊Q₈

558 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 116 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)(15,19)(17,24,18)(20,29,21,26,28,27)(22,25,23);(7,10)(11,13)(14,24,16,25)(15,29,19,28)(17,18,22,23)(20,27,26,21);(6,7)(8,11)(9,10)(12,13)(14,15,16,19)(17,20,22,26)(18,21,23,27)(24,28,25,29)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237055);