C₇×D₁₅₅ #14524244

order	2170
isoclinism	isoclinic class of D₁₅₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₅₅
unambiguous semidirect product	C₅⋊(C₇×D₃₁)
unambiguous semidirect product	C₃₁⋊(C₇×D₅)

Splittings

Fitting, socle, Hall	C₅×C₇×C₃₁ ⋊ C₂
derived, Hall	C₅×C₃₁ ⋊ C₂×C₇
Hall	C₇×C₃₁ ⋊ D₅
Hall	C₅×C₇ ⋊ D₃₁
Sylow	C₃₁ ⋊ C₇×D₅
Sylow	C₅ ⋊ C₇×D₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₅₅
commutator, cosocle, hypocenter	C₅×C₃₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₁	C₂
LCS	G ▷ C₅×C₃₁
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 1085

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₅×D₃₁, D₇×D₁₅₅, D₃₅⋊D₃₁, …
maximal normal subgroup of	C₇×D₅×D₃₁, D₇×D₁₅₅, D₃₅⋊D₃₁, …
maximal quotient of	C₇×Dic₁₅₅, C₂×C₇×D₁₅₅, C₇×D₇₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2170, 7);

C7×D155 #14524244