C₆₇×C₂²⋊(C₅²⋊C₃) #34117291

order	20100
isoclinism	isoclinic class of C₂²⋊(C₅²⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₆₇ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₆₇×C₂²⋊(C₅²⋊C₃)
unambiguous semidirect product	C₂²⋊(C₆₇×C₅²⋊C₃)
unambiguous semidirect product	C₅²⋊(A₄×C₆₇)

Splittings

Fitting, socle, Hall	C₂²×C₅²×C₆₇ ⋊ C₃
derived, Hall	C₂²×C₅² ⋊ C₃×C₆₇
Hall	C₅²×C₆₇ ⋊ A₄
Hall	C₂²×C₆₇ ⋊ C₅²⋊C₃
Sylow	C₅² ⋊ A₄×C₆₇
Sylow	C₂² ⋊ C₆₇×C₅²⋊C₃

Subgroups

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

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 2010

A minimal faithful permutation representation — 2 generators on 86 points:

(1,2,4)(3,6,8)(5,10,13)(7,12,14)(9,15,11)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86);(1,3,7,13,15)(2,6,12,5,11)(4,10,8,9,14)(16,19)(17,18)

Conjugacy classes & character table

2412 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 147);

C67×C22⋊(C52⋊C3) #34117291