C₅×C₁₁⋊D₃₄₁ #49176016

order	37510
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁₁

Aliases

direct product

C₅×C₁₁⋊D₃₄₁

Splittings

Fitting, socle, Hall	C₅×C₁₁²×C₃₁ ⋊ C₂
derived, Hall	C₁₁²×C₃₁ ⋊ C₂×C₅
Hall	C₅×C₁₁² ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ C₁₁⋊D₁₁
Sylow	C₁₁² ⋊ C₅×D₃₁
Sylow	C₃₁ ⋊ C₅×C₁₁⋊D₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₁₁⋊D₃₄₁
commutator, cosocle, hypocenter	C₁₁²×C₃₁	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁²×C₃₁	C₂
LCS	G ▷ C₁₁²×C₃₁
UCS	C₁ ◁ C₅

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 18755

A minimal faithful permutation representation — 3 generators on 58 points:

(1,2,3,4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(18,19)(20,21)(22,23)(24,25)(26,27)(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)(57,58);(17,18,20,22,24,26,27,25,23,21,19)(28,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30);(6,7,9,11,13,15,16,14,12,10,8)

Conjugacy classes & character table

9385 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37510, 178);

C5×C11⋊D341 #49176016