C₅²×Dic₃×C₆₇ #34117207

order	20100
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₅²×C₆₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×Dic₃×C₆₇
unambiguous semidirect product	C₃⋊(C₄×C₅²×C₆₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₆₇	D₃
commutator, hypocenter	C₃	C₄×C₅²×C₆₇
Frattini	C₂	C₅²×D₃×C₆₇
Fitting, socle, prime radical	C₂×C₃×C₅²×C₆₇	C₂
cosocle	C₂×C₃	C₂×C₅²×C₆₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅²×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 — 2 generators on 84 points:

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

Conjugacy classes & character table

10050 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 63);

C52×Dic3×C67 #34117207