C₅×#5124 #11750623

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

Aliases

direct product

C₅×#5124

Splittings

Hall	C₃×C₅ ⋊ #430
Sylow	C₃ ⋊ C₅×#430

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×#67
commutator	C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×M₁₆

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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)(9,11,12,20,14,21,13,22)(10,16,17,19,15,23,18,24);(10,15)(16,23)(17,18)(19,24);(7,8)(9,10,13,18,14,15,12,17)(11,19,20,23,21,24,22,16)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#70734, C₅×#70001, C₅×#72027, …
maximal normal subgroup of	C₅×#70734, C₅×#70001, C₅×#72027, …
maximal quotient of	C₅×#70420, C₅×#70746, C₅×#70492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4818);