C₅×#5698 #11756083

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

Aliases

direct product

C₅×#5698

Splittings

Fitting	C₃×C₅×#125 ⋊ C₂
Hall	C₃×C₅ ⋊ #541
Sylow	C₃ ⋊ C₅×#541

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#125

384 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,3,4,5)(6,7,8)(9,11,12,16)(10,14,13,15)(18,24)(20,29)(22,31)(25,28);(9,10,12,13)(11,15,16,14)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32);(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30);(7,8)(10,13)(14,15)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#81145, C₅×#81209, #14436257, …
maximal normal subgroup of	C₅×#81145, C₅×#81209, #14436257, …
maximal quotient of	C₅×#74224, C₅×#75970, C₅×#76197, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10278);