C₅×#3502⋊C₃ #14511351

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

solvable

Aliases

direct product	C₅×#3502⋊C₃
unambiguous semidirect product	#3502⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3502)⋊C₃

Splittings

Fitting, Hall	C₅×#3502 ⋊ C₃
Sylow	#3502 ⋊ 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×#3502	C₃
prime radical	C₂²×C₅×SL(2,𝔽₃)	C₂²
hypercenter	C₅×C₂²⋊Q₈	A₄
hypocenter	Q₈	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊Q₈

508 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 124 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)(15,19)(16,22,17)(20,29,21,26,28,27)(23,24,25);(6,8,9,13)(7,10,11,12)(14,24,18,22)(15,29,19,28)(16,17,23,25)(20,21,26,27);(6,7,9,11)(8,12,13,10)(14,15)(16,20)(17,21)(18,19)(22,28)(23,26)(24,29)(25,27)

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, 237049);