C₅×#4723 #11750222

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#4723

Splittings

Hall	C₃×C₅ ⋊ #485
Sylow	C₃ ⋊ C₅×#485

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₂²⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
hypercenter	C₅×C₂²⋊C₈	D₃
hypocenter	C₃	C₅×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×C₂²⋊C₈

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

normal subgroups only

This group has 192 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#485, C₂×C₅×#4723, D₅×#4723, …
maximal normal subgroup of	C₅×D₃×#485, C₂×C₅×#4723, D₅×#4723, …
maximal quotient of	C₅×#69926, C₅×#69929, C₅×#69931, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4417);