C₃×D₅⋊M₃₂ #11752248

order	960
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃×D₅⋊M₃₂
unambiguous semidirect product	D₅⋊(C₃×M₃₂)

Splittings

Hall	C₃×C₅ ⋊ C₂×M₃₂
Sylow	C₅ ⋊ C₂×C₃×M₃₂

Subgroups

	subgroup	G/H
center	C₃×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₃×C₈
Frattini	C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₄
socle	C₂×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₈
cosocle	#168	C₂³×C₃
hypercenter	C₂×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈

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

normal subgroups only

This group has 180 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)(9,11,13,20,14,12,19,21)(10,16,18,23,15,17,22,24);(10,15)(16,17)(18,22)(23,24);(5,6,7,8)(9,10,12,17,13,18,21,24,14,15,11,16,19,22,20,23)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598670, #14278732, C₃×#10598671, …
maximal normal subgroup of	C₃×#10598670, #14278732, C₃×#10598671, …
maximal quotient of	C₃×#10603823, C₃×#10604702, C₃×#10603825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6443);