C₅×#4690 #11750189

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#4690

Splittings

Fitting	C₃×C₄×C₅×C₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #433
Sylow	C₃ ⋊ C₅×#433

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂²×C₅×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈

192 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 94 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)(13,16)(17,19,20,25,21,26,27,31)(18,23,24,30,22,28,29,32);(9,11,12,15)(10,13,14,16)(18,22)(23,28)(24,29)(30,32);(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₃×#433, C₂×C₅×#4690, D₅×#4690, …
maximal normal subgroup of	C₅×D₃×#433, C₂×C₅×#4690, D₅×#4690, …
maximal quotient of	C₅×#69751, C₅×#69753, C₅×#69755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4384);