C₅×#69345 #14314590

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#69345

Splittings

Fitting	C₃×C₅×#367 ⋊ C₂
Hall	C₃×C₅ ⋊ #2158
Sylow	C₃ ⋊ C₅×#2158

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#367	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂²×C₅×D₃	D₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#367	D₃
hypocenter	C₃	C₅×#2158
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#367

304 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 44
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4,5)(6,7)(13,16,19,18)(14,21,23,33)(15,26,29,28)(17,30,32,25)(20,34,35,38)(22,31,36,44)(24,37,39,43)(27,41,42,40);(7,8)(13,15,25,40,18,28,32,42,19,29,30,41,16,26,17,27)(14,20,31,37,21,34,36,39,23,35,44,43,33,38,22,24);(9,10,11,12)(13,14)(15,24)(16,21)(17,31)(18,33)(19,23)(20,27)(22,25)(26,37)(28,43)(29,39)(30,36)(32,44)(34,41)(35,42)(38,40)

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 40288);