C₃×(C₅⋊D₅)⋊D₆₇ #34117259

order	20100
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×(C₅⋊D₅)⋊D₆₇

Splittings

Fitting, socle, Hall	C₃×C₅²×C₆₇ ⋊ C₂²
derived, Hall	C₅²×C₆₇ ⋊ C₂²×C₃
Hall	C₃×C₆₇ ⋊ D₅²
Hall	C₃×C₅² ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ C₃×D₅²
Sylow	C₅² ⋊ C₂×C₃×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	(C₅⋊D₅)⋊D₆₇
commutator, cosocle, hypocenter	C₅²×C₆₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₆₇	C₂²
LCS	G ▷ C₅²×C₆₇
UCS	C₁ ◁ C₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 2010

A minimal faithful permutation representation — 2 generators on 80 points:

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

Conjugacy classes & character table

1434 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 115);

C3×(C5⋊D5)⋊D67 #34117259