C₅×#5190 #11750689

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#5190

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #497
Sylow	C₃ ⋊ C₅×#497

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×D₁₂	C₂²
hypercenter	C₂×C₅×Q₁₆	D₃
hypocenter	C₃	C₅×#497
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₁₆

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

normal subgroups only

This group has 248 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)(10,12)(13,15,17,23)(14,19,20,28)(16,25,24,22)(18,26,27,21);(13,14,16,18,17,20,24,27)(15,21,22,28,23,26,25,19);(7,8)(9,10)(11,12)(14,18)(16,24)(19,26)(20,27)(21,28)(22,25)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#497, C₂×C₅×#5190, D₅×#5190, …
maximal normal subgroup of	C₅×D₃×#497, C₂×C₅×#5190, D₅×#5190, …
maximal quotient of	C₅×#71363, C₅×#71216, C₅×#71299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4884);