C₇×D₅×C₅₃ #16814977

order	3710
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₅×C₅₃
unambiguous semidirect product	C₅⋊(C₂×C₇×C₅₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₅₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₅₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₇×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);(9,12)(10,11)

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₇×D₅×C₅₃, C₇×D₁₅×C₅₃, D₅×ΓL(1,𝔽₈)×C₅₃, …
maximal normal subgroup of	C₃×C₇×D₅×C₅₃, D₅×ΓL(1,𝔽₈)×C₅₃, C₅×C₇×D₅×C₅₃, …
maximal quotient of	C₃×C₇×D₅×C₅₃, C₇×D₁₅×C₅₃, C₇×D₂₅×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3710, 5);

C7×D5×C53 #16814977