C₃×C₅×#1415 #14274449

order	1920
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×C₅×#1415

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃×C₅	D₁₆
commutator	C₁₆	C₂×C₃×C₄×C₅
Frattini, cosocle	C₂×C₁₆	C₂²×C₃×C₅
prime radical	C₂×C₃×C₅×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×C₈ ◁ C₂×C₃×C₅×C₁₆ ◁ G

164 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 2 generators on 44 points:

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

Conjugacy classes & character table

570 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 147);