C₆₇×C₃⋊#1039 #34117203

order	20100
isoclinism	isoclinic class of C₃⋊(C₅⋊D₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₆₇×C₃⋊#1039
unambiguous semidirect product	C₃⋊(C₆₇×#1039)

Splittings

derived, Hall	C₃×C₅² ⋊ C₄×C₆₇
Hall	C₃×C₅²×C₆₇ ⋊ C₄
Hall	C₅²×C₆₇ ⋊ Dic₃
Hall	C₃×C₆₇ ⋊ #1039
Sylow	C₅² ⋊ Dic₃×C₆₇
Sylow	C₃ ⋊ C₆₇×#1039

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₇	C₃⋊(C₅⋊D₅)
commutator, hypocenter	C₃×C₅²	C₄×C₆₇
Frattini	C₂	C₆₇×C₃⋊(C₅⋊D₅)
Fitting, socle, prime radical	C₂×C₃×C₅²×C₆₇	C₂
cosocle	C₂×C₃×C₅²	C₂×C₆₇
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₂×C₆₇

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 20100

A minimal faithful permutation representation — 3 generators on 84 points:

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

Conjugacy classes & character table

5226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 59);

C67×C3⋊#1039 #34117203