C₇×D₈₅ #11924735

order	1190
isoclinism	isoclinic class of D₈₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₈₅
unambiguous semidirect product	C₅⋊(C₇×D₁₇)
unambiguous semidirect product	C₁₇⋊(C₇×D₅)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₇ ⋊ C₂
derived, Hall	C₅×C₁₇ ⋊ C₂×C₇
Hall	C₇×C₁₇ ⋊ D₅
Hall	C₅×C₇ ⋊ D₁₇
Sylow	C₁₇ ⋊ C₇×D₅
Sylow	C₅ ⋊ C₇×D₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₈₅
commutator, cosocle, hypocenter	C₅×C₁₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₇	C₂
LCS	G ▷ C₅×C₁₇
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 595

A minimal faithful permutation representation — 2 generators on 29 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);(9,12)(10,11)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×#66775, C₇×#66776, C₇⋊#66775, …
maximal normal subgroup of	C₇×#66775, C₇×#66776, C₇⋊#66775, …
maximal quotient of	C₇×Dic₈₅, C₂×C₇×D₈₅, C₃×C₇×D₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1190, 5);

C7×D85 #11924735