C₇×#5812 #13061525

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₇×#5812

Splittings

Hall	C₃×C₇ ⋊ #519
Sylow	C₃ ⋊ C₇×#519

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂²×C₃	C₂⁴×C₇
hypercenter	C₂²×C₇×Q₈	D₃
hypocenter	C₃	C₇×#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×Q₈

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10814);

C7×#5812 #13061525