C₅×#4828 #11750327

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#4828

Splittings

Hall	C₃×C₅ ⋊ #446
Sylow	C₃ ⋊ C₅×#446

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×C₄⋊C₈	D₃
hypocenter	C₃	C₅×#446
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊C₈

188 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 960

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

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#446, C₂×C₅×#4828, D₅×#4828, …
maximal normal subgroup of	C₅×D₃×#446, C₂×C₅×#4828, D₅×#4828, …
maximal quotient of	C₅×#68652, C₅×#68654, C₅×#70467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4522);