C₅×#5089 #11750588

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

Aliases

direct product

C₅×#5089

Splittings

Fitting	C₃×C₅×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #489
Sylow	C₃ ⋊ C₅×#489

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#489
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₄⋊Q₈

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 480

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

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71329, C₅×#71244, #14291011, …
maximal normal subgroup of	C₅×#71329, C₅×#71244, #14291011, …
maximal quotient of	C₅×#69575, C₅×#69616, C₅×#69617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4783);