C₅×C₇⋊D₄₆₉ #45659182

order	32830
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₅×C₇⋊D₄₆₉
unambiguous semidirect product	C₇⋊(C₅×D₄₆₉)
unambiguous semidirect product	C₆₇⋊(C₅×C₇⋊D₇)

Splittings

Fitting, socle, Hall	C₅×C₇²×C₆₇ ⋊ C₂
derived, Hall	C₇²×C₆₇ ⋊ C₂×C₅
Hall	C₅×C₆₇ ⋊ C₇⋊D₇
Hall	C₅×C₇² ⋊ D₆₇
Sylow	C₆₇ ⋊ C₅×C₇⋊D₇
Sylow	C₇² ⋊ C₅×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₇⋊D₄₆₉
commutator, cosocle, hypocenter	C₇²×C₆₇	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₆₇	C₂
LCS	G ▷ C₇²×C₆₇
UCS	C₁ ◁ C₅

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 16415

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

Conjugacy classes & character table

8215 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32830, 14);

C5×C7⋊D469 #45659182