C₇×#4997 #13056031

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#4997

Splittings

Hall	C₃×C₇ ⋊ #405
Sylow	C₃ ⋊ C₇×#405

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#117

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5,6,7)(8,9,10)(11,12,13,14)(17,23)(20,26)(24,29)(27,30);(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(9,10)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30)

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, 5320);

C7×#4997 #13056031