C₇×D₁₇₁ #14558796

order	2394
isoclinism	isoclinic class of D₁₇₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₇₁
unambiguous semidirect product	C₉⋊(C₇×D₁₉)
unambiguous semidirect product	C₁₉⋊(C₇×D₉)

Splittings

Fitting, Hall	C₇×C₉×C₁₉ ⋊ C₂
derived, Hall	C₉×C₁₉ ⋊ C₂×C₇
Hall	C₇×C₁₉ ⋊ D₉
Hall	C₇×C₉ ⋊ D₁₉
Sylow	C₁₉ ⋊ C₇×D₉
Sylow	C₉ ⋊ C₇×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₇₁
commutator, cosocle, hypocenter	C₉×C₁₉	C₂×C₇
Frattini	C₃	C₇×D₅₇
Fitting	C₇×C₉×C₁₉	C₂
socle	C₃×C₇×C₁₉	D₃
LCS	G ▷ C₉×C₁₉
UCS	C₁ ◁ C₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1197

A minimal faithful permutation representation — 2 generators on 35 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);(9,16)(10,15)(11,14)(12,13)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₉×D₁₉, D₇×D₁₇₁, D₆₃⋊D₁₉, …
maximal normal subgroup of	C₇×D₉×D₁₉, D₇×D₁₇₁, D₆₃⋊D₁₉, …
maximal quotient of	C₇×Dic₁₇₁, C₂×C₇×D₁₇₁, C₅×C₇×D₁₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2394, 22);

C7×D171 #14558796