C₄₁×C₅⋊(C₇⋊D₇) #34095606

order	20090
isoclinism	isoclinic class of C₅⋊(C₇⋊D₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄₁ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₄₁×C₅⋊(C₇⋊D₇)
unambiguous semidirect product	C₅⋊(C₄₁×C₇⋊D₇)
unambiguous semidirect product	C₇⋊(C₄₁×D₃₅)

Splittings

Fitting, socle, Hall	C₅×C₇²×C₄₁ ⋊ C₂
derived, Hall	C₅×C₇² ⋊ C₂×C₄₁
Hall	C₇²×C₄₁ ⋊ D₅
Hall	C₅×C₄₁ ⋊ C₇⋊D₇
Sylow	C₇² ⋊ D₅×C₄₁
Sylow	C₅ ⋊ C₄₁×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	C₅⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₅×C₇²	C₂×C₄₁
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₄₁	C₂
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₄₁

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 10045

A minimal faithful permutation representation — 3 generators on 60 points:

(2,3)(4,5)(7,8)(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);(1,2,4,5,3)(13,14,16,18,19,17,15);(6,7,9,11,12,10,8)

Conjugacy classes & character table

5084 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20090, 29);

C41×C5⋊(C7⋊D7) #34095606