C₅×C₇×C₄₃⋊C₃ #17985259

order	4515
isoclinism	isoclinic class of C₄₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×C₇×C₄₃⋊C₃

Splittings

Fitting, socle, Hall	C₅×C₇×C₄₃ ⋊ C₃
derived, Sylow	C₄₃ ⋊ C₃×C₅×C₇
Hall	C₇×C₄₃ ⋊ C₃×C₅
Hall	C₅×C₄₃ ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₇	C₄₃⋊C₃
commutator, cosocle, hypocenter	C₄₃	C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₄₃	C₃
LCS	G ▷ C₄₃
UCS	C₁ ◁ C₅×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1505

A minimal faithful permutation representation — 2 generators on 55 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×ΓL(1,𝔽₈)×C₄₃⋊C₃, C₃×C₅×C₇×C₄₃⋊C₃, C₅×C₇×C₂²⋊(C₄₃⋊C₃), …
maximal normal subgroup of	C₅×ΓL(1,𝔽₈)×C₄₃⋊C₃, C₃×C₅×C₇×C₄₃⋊C₃, C₇×C₂₅×C₄₃⋊C₃, …
maximal quotient of	C₅×C₇×C₄₃⋊C₉, C₃×C₅×C₇×C₄₃⋊C₃, C₅×C₇×C₂²⋊(C₄₃⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4515, 3);

C5×C7×C43⋊C3 #17985259