C₇×C₅⋊D₃₆₅ #38784382

order	25550
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₇×C₅⋊D₃₆₅
unambiguous semidirect product	C₅⋊(C₇×D₃₆₅)
unambiguous semidirect product	C₇₃⋊(C₇×C₅⋊D₅)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₇₃ ⋊ C₂
derived, Hall	C₅²×C₇₃ ⋊ C₂×C₇
Hall	C₇×C₇₃ ⋊ C₅⋊D₅
Hall	C₅²×C₇ ⋊ D₇₃
Sylow	C₇₃ ⋊ C₇×C₅⋊D₅
Sylow	C₅² ⋊ C₇×D₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₅⋊D₃₆₅
commutator, cosocle, hypocenter	C₅²×C₇₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₇₃	C₂
LCS	G ▷ C₅²×C₇₃
UCS	C₁ ◁ C₇

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 12775

A minimal faithful permutation representation — 3 generators on 90 points:

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

Conjugacy classes & character table

6398 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25550, 18);

C7×C5⋊D365 #38784382