C₅×#67829 #14279602

order	1920
isoclinism	isoclinic class of #4519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67829

Splittings

Hall	C₃×C₅ ⋊ #1417
Sylow	C₃ ⋊ C₅×#1417

Subgroups

	subgroup	G/H
center	C₄×C₅	#826
commutator	C₃×C₁₆	C₂×C₄×C₅
Frattini	C₂×C₁₆	C₂×C₅×D₃
Fitting, prime radical	C₃×C₅×#508	C₂
socle	C₂×C₃×C₅	#357
cosocle	C₂×C₃×C₁₆	C₂²×C₅
hypercenter	C₅×#508	D₃
hypocenter	C₃	C₅×#1417
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₅×#508

144 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 72
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,4,5)(6,7,8)(9,11)(10,16)(12,29)(13,24)(14,36)(15,27)(17,43)(18,39)(19,48)(20,50)(21,41)(22,28)(23,32)(25,30)(26,34)(31,62)(33,64)(35,40)(37,55)(38,42)(44,56)(45,71)(46,70)(47,68)(49,72)(51,59)(52,61)(53,69)(54,67)(57,65)(58,60)(63,66);(7,8)(9,10,13,18,15,21,34,20)(11,22,24,51,27,52,26,58)(12,28,31,59,33,61,66,60)(14,35,30,19,37,47,65,49)(16,38,39,54,41,53,50,23)(17,42,44,67,46,69,45,32)(25,56,55,70,57,71,36,43)(29,48,62,68,64,72,63,40)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5300);