C₅×#3523⋊C₃ #14511358

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

solvable

Aliases

direct product	C₅×#3523⋊C₃
unambiguous semidirect product	#3523⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3523)⋊C₃

Splittings

Fitting, Hall	C₅×#3523 ⋊ C₃
Sylow	#3523 ⋊ 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×#3523	C₃
hypercenter	C₅×#121	A₄
hypocenter	Q₈	C₃×C₅×#41
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#121

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

normal subgroups only

This group has 554 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)(17,23,25)(18,24,22)(20,21,29,26,27,28);(7,10)(11,13)(14,25,16,24)(15,28,19,29)(17,18,22,23)(20,27,26,21);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,20)(18,21)(22,26)(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, 237056);