C₅×#67459 #14278307

order	1920
isoclinism	isoclinic class of #67448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67459

Splittings

Hall	C₃×C₅ ⋊ #1353
Sylow	C₃ ⋊ C₅×#1353

Subgroups

	subgroup	G/H
center	C₂²×C₅	#806
commutator	C₂²×C₃×C₄	C₂×C₄×C₅
Frattini	C₂×C₄⋊C₄	C₂×C₅×D₃
Fitting	C₃×C₅×#324	C₂
socle	C₂²×C₃×C₅	#99
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₂×C₃×C₄⋊C₄	C₂²×C₅
hypercenter	C₅×#324	D₃
hypocenter	C₃	C₅×#1353
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ C₅×#324

232 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,4,5)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22,25,32)(20,28)(21,29)(23,31,24,26)(27,30);(7,8)(10,13,14,15)(17,18,20,24,21,25,30,23)(19,26,28,22,29,31,27,32)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 4005);