C₅×C₇⋊D₅₁₁ #49088709

order	35770
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₅×C₇⋊D₅₁₁
unambiguous semidirect product	C₇⋊(C₅×D₅₁₁)
unambiguous semidirect product	C₇₃⋊(C₅×C₇⋊D₇)

Splittings

Fitting, socle, Hall	C₅×C₇²×C₇₃ ⋊ C₂
derived, Hall	C₇²×C₇₃ ⋊ C₂×C₅
Hall	C₅×C₇₃ ⋊ C₇⋊D₇
Hall	C₅×C₇² ⋊ D₇₃
Sylow	C₇₃ ⋊ C₅×C₇⋊D₇
Sylow	C₇² ⋊ C₅×D₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₇⋊D₅₁₁
commutator, cosocle, hypocenter	C₇²×C₇₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₇₃	C₂
LCS	G ▷ C₇²×C₇₃
UCS	C₁ ◁ C₅

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 17885

A minimal faithful permutation representation — 3 generators on 92 points:

(1,2,3,4,5)(7,8)(9,10)(11,12)(14,15)(16,17)(18,19)(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)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92);(13,14,16,18,19,17,15)(20,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22);(6,7,9,11,12,10,8)

Conjugacy classes & character table

8950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35770, 14);

C5×C7⋊D511 #49088709