C₇×#4916 #13055951

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

Aliases

direct product

C₇×#4916

Splittings

Fitting	C₃×C₇×SD₃₂ ⋊ 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×SD₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
hypercenter	C₇×SD₃₂	D₃
hypocenter	C₃	C₇×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×SD₃₂

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

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, 5240);