C₅×#5734 #11756119

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#5734

Splittings

Fitting	C₃×C₅×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #555
Sylow	C₃ ⋊ C₅×#555

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊Q₈

472 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 24 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71265, C₅×#68491, #14289066, …
maximal normal subgroup of	C₅×#71265, C₅×#68491, #14289066, …
maximal quotient of	C₅×#75979, C₅×#74324, C₅×#75985, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10314);