C₂×C₆₇×(C₅⋊D₅)⋊C₃ #34117243

order	20100
isoclinism	isoclinic class of (C₅⋊D₅)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₆₇ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₂×C₆₇×(C₅⋊D₅)⋊C₃
unambiguous semidirect product	(C₂×C₅⋊D₅)⋊(C₃×C₆₇)

Splittings

Fitting, socle	C₂×C₅²×C₆₇ ⋊ C₂×C₃
derived, Sylow	C₅² ⋊ C₂²×C₃×C₆₇
Hall	C₂×C₅⋊D₅×C₆₇ ⋊ C₃
Hall	C₆₇×C₅²⋊C₃ ⋊ C₂²
Hall	C₅²×C₆₇ ⋊ C₂²×C₃
Hall	C₂×C₅⋊D₅ ⋊ C₃×C₆₇
Hall	C₅²⋊C₃ ⋊ C₂²×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₇	(C₅⋊D₅)⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂²×C₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₆₇	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₆₇

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 2010

A minimal faithful permutation representation — 2 generators on 84 points:

(1,2,4)(3,6,10,7,5,12)(8,13,15,14,11,9)(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);(1,3,8,14,7)(2,5,11,13,6)(4,9,10,12,15)(83,84)

Conjugacy classes & character table

1340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 99);

C2×C67×(C5⋊D5)⋊C3 #34117243