C₅×Dic₈₇ #13181950

order	1740
isoclinism	isoclinic class of D₈₇
automorphism	Aut(G) = C₂×C₄×D₃×AGL(1,𝔽₂₉) (order 38976) · Inn(G) = D₈₇ · Out(G) = C₂×C₄²×C₇
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Dic₈₇
unambiguous semidirect product	C₃⋊(C₅×Dic₂₉)

Splittings

derived, Hall	C₃×C₂₉ ⋊ C₄×C₅
Hall	C₃×C₅×C₂₉ ⋊ C₄
Hall	C₅×C₂₉ ⋊ Dic₃
Sylow	C₂₉ ⋊ C₅×Dic₃
Hall	C₃×C₅ ⋊ Dic₂₉
Sylow	C₃ ⋊ C₅×Dic₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₈₇
commutator, hypocenter	C₃×C₂₉	C₄×C₅
Frattini	C₂	C₅×D₈₇
Fitting, socle, prime radical	C₂×C₃×C₅×C₂₉	C₂
cosocle	C₂×C₃×C₂₉	C₂×C₅
LCS	G ▷ C₃×C₂₉
UCS	C₁ ◁ C₂×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 1740

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(6,8,7,9)(11,12)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₁₁×Dic₈₇, C₅×Dic₉₅₇, C₁₁⋊C₅×Dic₈₇, …
maximal normal subgroup of	C₅×C₁₁×Dic₈₇, C₅×C₁₃×Dic₈₇, C₅×C₁₇×Dic₈₇, …
maximal quotient of	C₅×C₁₁×Dic₈₇, C₅×Dic₉₅₇, C₁₁⋊C₅×Dic₈₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1740, 5);

C5×Dic87 #13181950