C₇×#4919 #13055954

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4919

Splittings

Fitting	C₃×C₇×Q₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #508
Sylow	C₃ ⋊ C₇×#508

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×Q₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
hypercenter	C₇×Q₃₂	D₃
hypocenter	C₃	C₇×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×Q₃₂

168 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 3 generators on 42 points:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5243);