C₇×#5753 #13061469

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#5753

Splittings

Fitting	C₃×C₇×#131 ⋊ C₂
Hall	C₃×C₇ ⋊ #578
Sylow	C₃ ⋊ C₇×#578

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ C₇×#131

496 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 42 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10758);