C₅×D₄₅₁ #17985228

order	4510
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×D₄₅₁

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₄₁ ⋊ C₂
derived, Hall	C₁₁×C₄₁ ⋊ C₂×C₅
Hall	C₅×C₄₁ ⋊ D₁₁
Hall	C₅×C₁₁ ⋊ D₄₁
Sylow	C₄₁ ⋊ C₅×D₁₁
Sylow	C₁₁ ⋊ C₅×D₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₄₅₁
commutator, cosocle, hypocenter	C₁₁×C₄₁	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₄₁	C₂
LCS	G ▷ C₁₁×C₄₁
UCS	C₁ ◁ C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2255

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(7,16)(8,15)(9,14)(10,13)(11,12)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Conjugacy classes & character table

1135 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅×D₄₅₁, C₅×D₁₃₅₃, C₂₅×D₄₅₁, …
maximal normal subgroup of	C₃×C₅×D₄₅₁, C₂₅×D₄₅₁, (C₄₁⋊C₂₅)⋊D₁₁, …
maximal quotient of	C₃×C₅×D₄₅₁, C₅×D₁₃₅₃, C₂₅×D₄₅₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4510, 10);

C5×D451 #17985228