C₅×#5129 #11750628

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#5129

Splittings

Fitting	C₂×C₃×C₅×M₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #408
Sylow	C₃ ⋊ C₅×#408

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂×C₅×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#408
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆

316 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 126 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)(10,12)(13,15,16,22,17,21,23,27)(14,19,20,26,18,24,25,28);(14,18)(15,21)(20,25)(22,27);(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67531, #14277633, C₅×#70143, …
maximal normal subgroup of	C₅×#67531, #14277633, C₅×#70143, …
maximal quotient of	C₅×#69969, C₅×#70828, C₅×#70831, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4823);