C₇×#5470 #13061186

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#5470

Splittings

Hall	C₃×C₇ ⋊ #533
Sylow	C₃ ⋊ C₇×#533

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₄²×C₇	D₃
hypocenter	C₃	C₇×#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10475);

C7×#5470 #13061186