C₇²×D₅×C₄₁ #34095602

order	20090
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇²×C₄₁ · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×D₅×C₄₁
unambiguous semidirect product	C₅⋊(C₂×C₇²×C₄₁)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₇²×C₄₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₇²×C₄₁
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₄₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₇²×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 — 2 generators on 60 points:

(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)(54,55,56,57,58,59,60);(1,2,4,5,3)(47,48,49,50,51,52,53)

Conjugacy classes & character table

8036 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20090, 25);

C72×D5×C41 #34095602