C₃×C₅×#1315 #14274349

order	1920
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₅×#1315

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂²×C₃×C₅
socle	C₂²×C₃×C₅	C₂²⋊C₈
prime radical	C₂×C₃×C₅×#41	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

232 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 48
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,3)(4,5,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,31,46,30,40,47,43,32,41,35,48,33,24,38,39);(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,41)(26,43)(27,29)(28,40)(31,37)(34,47)(36,38)(39,45)(42,46)(44,48)

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, 47);