C₅×#5177 #11750676

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#5177

Splittings

Fitting	C₂×C₃×C₅×SD₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #471
Sylow	C₃ ⋊ C₅×#471

Subgroups

	subgroup	G/H
center	C₂×C₅	#938
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×SD₁₆	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₅×#932	C₂
hypercenter	C₂×C₅×SD₁₆	D₃
hypocenter	C₃	C₅×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×SD₁₆

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(10,15)(11,18)(12,20)(14,21)(16,17)(19,24);(9,11,12,18,13,19,20,24)(10,14,15,23,17,21,16,22);(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,23)(19,21)(22,24)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#471, C₂×C₅×#5177, D₅×#5177, …
maximal normal subgroup of	C₅×D₃×#471, C₂×C₅×#5177, D₅×#5177, …
maximal quotient of	C₅×#71267, C₅×#71272, C₅×#71344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4871);