C₇×#5140 #13056175

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#5140

Splittings

Hall	C₃×C₇ ⋊ #504
Sylow	C₃ ⋊ C₇×#504

Subgroups

	subgroup	G/H
center	C₇×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	C₂²×C₈
prime radical	C₃×C₇×#41	C₄
cosocle	C₃×C₈	C₂³×C₇
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₇×#131

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(13,21)(17,27)(22,33)(23,34)(28,37)(29,38)(35,41)(39,42);(11,13)(12,17)(14,22)(15,23)(16,21)(18,28)(19,29)(20,27)(24,35)(25,33)(26,34)(30,39)(31,37)(32,38)(36,41)(40,42);(9,10)(11,12,14,18,15,19,24,30,16,20,25,31,26,32,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42)

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, 5464);

C7×#5140 #13056175