C₅₃×D₃₅ #16814979

order	3710
isoclinism	isoclinic class of D₃₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₃₅
unambiguous semidirect product	C₅⋊(D₇×C₅₃)
unambiguous semidirect product	C₇⋊(D₅×C₅₃)

Splittings

Fitting, socle, Hall	C₅×C₇×C₅₃ ⋊ C₂
derived, Hall	C₅×C₇ ⋊ C₂×C₅₃
Hall	C₇×C₅₃ ⋊ D₅
Hall	C₅×C₅₃ ⋊ D₇
Sylow	C₇ ⋊ D₅×C₅₃
Sylow	C₅ ⋊ D₇×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₃₅
commutator, cosocle, hypocenter	C₅×C₇	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₅₃	C₂
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₅₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1855

A minimal faithful permutation representation — 2 generators on 65 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55,56,57,58)(59,60,61,62,63,64,65);(55,58)(56,57)(60,65)(61,64)(62,63)

Conjugacy classes & character table

1007 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅₃×D₃₅, C₅₃×D₁₀₅, C₅₃×C₅⋊AGL(1,𝔽₇), …
maximal normal subgroup of	C₃×C₅₃×D₃₅, C₅₃×C₅⋊AGL(1,𝔽₇), C₅×C₅₃×D₃₅, …
maximal quotient of	C₃×C₅₃×D₃₅, C₅₃×D₁₀₅, C₅₃×D₁₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3710, 7);

C53×D35 #16814979