C₇×D₁₈₅ #14579554

order	2590
isoclinism	isoclinic class of D₁₈₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₈₅
unambiguous semidirect product	C₅⋊(C₇×D₃₇)
unambiguous semidirect product	C₃₇⋊(C₇×D₅)

Splittings

Fitting, socle, Hall	C₅×C₇×C₃₇ ⋊ C₂
derived, Hall	C₅×C₃₇ ⋊ C₂×C₇
Hall	C₇×C₃₇ ⋊ D₅
Sylow	C₃₇ ⋊ C₇×D₅
Hall	C₅×C₇ ⋊ D₃₇
Sylow	C₅ ⋊ C₇×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₈₅
commutator, cosocle, hypocenter	C₅×C₃₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₇	C₂
LCS	G ▷ C₅×C₃₇
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1295

A minimal faithful permutation representation — 2 generators on 49 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);(9,12)(10,11)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₉₂₅, C₇×C₅⋊D₁₈₅, C₅×C₇×D₁₈₅, …
maximal normal subgroup of	C₅×C₇×D₁₈₅, C₄₉×D₁₈₅, C₇²×D₁₈₅, …
maximal quotient of	C₇×D₉₂₅, C₇×C₅⋊D₁₈₅, C₅×C₇×D₁₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2590, 6);

C7×D185 #14579554