C₅×D₇×C₅₃ #16814975

order	3710
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₇×C₅₃
unambiguous semidirect product	C₇⋊(C₂×C₅×C₅₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₅₃	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₅×C₅₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₅₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₅×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);(7,12)(8,11)(9,10)

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅×D₇×C₅₃, C₅×D₂₁×C₅₃, C₅×AGL(1,𝔽₇)×C₅₃, …
maximal normal subgroup of	C₃×C₅×D₇×C₅₃, C₅×AGL(1,𝔽₇)×C₅₃, D₇×C₂₅×C₅₃, …
maximal quotient of	C₃×C₅×D₇×C₅₃, C₅×D₂₁×C₅₃, D₇×C₂₅×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3710, 3);

C5×D7×C53 #16814975