C₃×D₅⋊#108 #11752257

order	960
isoclinism	isoclinic class of #3993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃×D₅⋊#108
unambiguous semidirect product	D₅⋊(C₃×#108)

Splittings

Hall	C₃×C₅ ⋊ C₂×#108
Sylow	C₅ ⋊ C₂×C₃×#108

Subgroups

	subgroup	G/H
center	C₃×C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₄
Frattini	C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₄
socle	C₂×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	Q₈
cosocle	C₄×D₅	C₂³×C₃
hypercenter	C₂×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

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

normal subgroups only

This group has 212 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)(9,12,13,20,14,21,11,18)(10,16,15,19,17,23,22,24);(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,19,23,24);(5,6,7,8)(9,10,11,15,14,17,13,22)(12,19,18,23,21,24,20,16)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598625, #14278846, C₃×#10598665, …
maximal normal subgroup of	C₃×#10598625, #14278846, C₃×#10598665, …
maximal quotient of	C₃×D₅⋊#334, C₃×D₅⋊#335, C₃×#10604694, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6452);