C₇×C₅⋊D₅₀₅ #47977840

order	35350
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₇×C₅⋊D₅₀₅
unambiguous semidirect product	C₅⋊(C₇×D₅₀₅)
unambiguous semidirect product	C₁₀₁⋊(C₇×C₅⋊D₅)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₁₀₁ ⋊ C₂
derived, Hall	C₅²×C₁₀₁ ⋊ C₂×C₇
Hall	C₇×C₁₀₁ ⋊ C₅⋊D₅
Hall	C₅²×C₇ ⋊ D₁₀₁
Sylow	C₁₀₁ ⋊ C₇×C₅⋊D₅
Sylow	C₅² ⋊ C₇×D₁₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₅⋊D₅₀₅
commutator, cosocle, hypocenter	C₅²×C₁₀₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₁₀₁	C₂
LCS	G ▷ C₅²×C₁₀₁
UCS	C₁ ◁ C₇

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 17675

A minimal faithful permutation representation — 3 generators on 118 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(14,15)(16,17)(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)(57,58)(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)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118);(13,14,16,17,15)(18,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20);(8,9,11,12,10)

Conjugacy classes & character table

8848 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35350, 34);

C7×C5⋊D505 #47977840