C₅×C₄₁×C₇⋊D₇ #34095600

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

Aliases

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

Splittings

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

Subgroups

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

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)(43,44)(45,46)(47,48)(50,51)(52,53)(54,55)(56,57,58,59,60);(49,50,52,54,55,53,51);(42,43,45,47,48,46,44)

Conjugacy classes & character table

5330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20090, 23);

C5×C41×C7⋊D7 #34095600