C₅×#5003 #11750502

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

Aliases

direct product

C₅×#5003

Splittings

Fitting	C₃×C₅×#117 ⋊ C₂
Hall	C₃×C₅ ⋊ #482
Sylow	C₃ ⋊ C₅×#482

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×#117	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₅×#930	C₂
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#482
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#117

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

normal subgroups only

This group has 212 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)(9,11,12,15)(10,14,13,16)(17,19,21,23)(18,22,20,24);(10,13)(14,16)(17,18,21,20)(19,22,23,24);(7,8)(9,10)(11,14)(12,13)(15,16)(18,20)(19,22)(23,24)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#482, C₂×C₅×#5003, D₅×#5003, …
maximal normal subgroup of	C₅×D₃×#482, C₂×C₅×#5003, D₅×#5003, …
maximal quotient of	C₅×#71575, C₅×#71576, C₅×#71586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4697);