C₇×#6129 #13089887

order	1456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#6129

Splittings

Fitting	C₇×D₄×C₁₃ ⋊ C₂
Hall	C₇×C₁₃ ⋊ #41
Sylow	C₁₃ ⋊ C₇×#41

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×D₁₃
commutator, cosocle	C₂×C₁₃	C₂³×C₇
Frattini	C₂	C₂²×C₇×D₁₃
Fitting	C₇×D₄×C₁₃	C₂
socle	C₂×C₇×C₁₃	C₂³
hypercenter	C₇×D₄	D₁₃
hypocenter	C₁₃	C₇×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×D₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 728

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5,6,7)(8,9,11,13,15,17,19,20,18,16,14,12,10)(22,26)(23,27);(21,23,24,27)(22,25,26,28);(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,25)(24,26)(27,28)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1456, 141);

C7×#6129 #13089887