C₅×#5187 #11750686

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

Aliases

direct product

C₅×#5187

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #452
Sylow	C₃ ⋊ C₅×#452

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×Q₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#239	C₂
hypercenter	C₂×C₅×Q₁₆	D₃
hypocenter	C₃	C₅×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₁₆

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 28
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#452, C₂×C₅×#5187, D₅×#5187, …
maximal normal subgroup of	C₅×D₃×#452, C₂×C₅×#5187, D₅×#5187, …
maximal quotient of	C₅×#70982, C₅×#70981, C₅×#71178, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4881);