D₆₇×C₃⋊(C₅⋊D₅) #34117282

order	20100
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	D₆₇×C₃⋊(C₅⋊D₅)
unambiguous semidirect product	(C₃×D₆₇)⋊(C₅⋊D₅)
unambiguous semidirect product	(C₅×D₆₇)⋊D₁₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅²×C₆₇ ⋊ C₂²
Hall	C₅²×C₆₇ ⋊ C₂×D₃
Hall	C₃×C₆₇ ⋊ C₂×C₅⋊D₅
Hall	C₃×C₅² ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ C₂×C₃⋊(C₅⋊D₅)
Sylow	C₅² ⋊ D₃×D₆₇
Sylow	C₃ ⋊ C₅⋊D₅×D₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₆₇	C₂²
LCS	G ▷ C₃×C₅²×C₆₇
UCS	C₁

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 5025

A minimal faithful permutation representation — 3 generators on 80 points:

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

Conjugacy classes & character table

1365 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 138);

D67×C3⋊(C5⋊D5) #34117282