C₅×#5095 #11750594

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

Aliases

direct product

C₅×#5095

Splittings

Fitting	C₃×C₅×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #456
Sylow	C₃ ⋊ C₅×#456

Subgroups

	subgroup	G/H
center	C₂×C₅	#937
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊Q₈	C₂
socle	C₂×C₃×C₅	#120
prime radical	C₅×#1007	C₂
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#456
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊Q₈

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

normal subgroups only

This group has 284 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) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67550, C₅×#67463, #14279378, …
maximal normal subgroup of	C₅×#67550, C₅×#67463, #14279378, …
maximal quotient of	C₅×#69705, C₅×#69706, C₅×#69707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4789);