C₇×D₈₇ #11927396

order	1218
isoclinism	isoclinic class of D₈₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₈₇
unambiguous semidirect product	C₃⋊(C₇×D₂₉)

Splittings

Fitting, socle, Hall	C₃×C₇×C₂₉ ⋊ C₂
derived, Hall	C₃×C₂₉ ⋊ C₂×C₇
Hall	C₇×C₂₉ ⋊ D₃
Sylow	C₂₉ ⋊ D₃×C₇
Hall	C₃×C₇ ⋊ D₂₉
Sylow	C₃ ⋊ C₇×D₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₈₇
commutator, cosocle, hypocenter	C₃×C₂₉	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₂₉	C₂
LCS	G ▷ C₃×C₂₉
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 609

A minimal faithful permutation representation — 2 generators on 39 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);(9,10)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃×C₇×D₂₉, D₇×D₈₇, D₂₁⋊D₂₉, …
maximal normal subgroup of	D₃×C₇×D₂₉, D₇×D₈₇, D₂₁⋊D₂₉, …
maximal quotient of	C₇×Dic₈₇, C₂×C₇×D₈₇, C₇×D₂₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1218, 13);

C7×D87 #11927396