C₃×D₅⋊#101 #11752322

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₃×D₅⋊#101
unambiguous semidirect product	D₅⋊(C₃×#101)

Splittings

Hall	C₃×C₅ ⋊ C₂×#101
Sylow	C₅ ⋊ C₂×C₃×#101

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×Q₈	C₄
socle	C₂×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#101
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₈

292 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 100 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)(4,5,6,8,7)(9,12,14,22)(10,16,18,20)(11,19,13,21)(15,23,17,24);(9,11,14,13)(10,15,18,17)(12,21,22,19)(16,24,20,23);(5,6,7,8)(9,10,13,17,14,18,11,15)(12,20,19,24,22,16,21,23)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598577, #14279146, C₃×AGL(1,𝔽₅)×#101, …
maximal normal subgroup of	C₃×#10598577, #14279146, C₃×AGL(1,𝔽₅)×#101, …
maximal quotient of	C₃×D₅⋊#324, C₃×#10604976, C₃×#10604291, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6517);