C₃×C₆₇×#1042 #34117255

order	20100
isoclinism	isoclinic class of #1042
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₆₇×#1042

Splittings

Fitting, socle, Hall	C₃×C₅²×C₆₇ ⋊ C₄
derived, Sylow	C₅² ⋊ C₃×C₄×C₆₇
Hall	C₅²×C₆₇ ⋊ C₃×C₄
Hall	C₃×C₅² ⋊ C₄×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₆₇	#1042
commutator, hypocenter	C₅²	C₃×C₄×C₆₇
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₆₇	C₄
prime radical	C₃×C₅×D₅×C₆₇	C₂
cosocle	C₅×D₅	C₂×C₃×C₆₇
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃×C₆₇

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 4020

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

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

Conjugacy classes & character table

2613 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 111);

C3×C67×#1042 #34117255