C₇×D₁₁₅ #13108012

order	1610
isoclinism	isoclinic class of D₁₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₁₅
unambiguous semidirect product	C₅⋊(C₇×D₂₃)
unambiguous semidirect product	C₂₃⋊(C₇×D₅)

Splittings

Fitting, socle, Hall	C₅×C₇×C₂₃ ⋊ C₂
derived, Hall	C₅×C₂₃ ⋊ C₂×C₇
Hall	C₇×C₂₃ ⋊ D₅
Hall	C₅×C₇ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₇×D₅
Sylow	C₅ ⋊ C₇×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₁₅
commutator, cosocle, hypocenter	C₅×C₂₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₂₃	C₂
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 805

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);(9,12)(10,11)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₅×D₂₃, D₇×D₁₁₅, D₃₅⋊D₂₃, …
maximal normal subgroup of	C₇×D₅×D₂₃, D₇×D₁₁₅, D₃₅⋊D₂₃, …
maximal quotient of	C₇×Dic₁₁₅, C₂×C₇×D₁₁₅, C₃×C₇×D₁₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1610, 5);

C7×D115 #13108012