C₇×#5204 #13056237

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

Aliases

direct product

C₇×#5204

Splittings

Hall	C₃×C₇ ⋊ #473
Sylow	C₃ ⋊ C₇×#473

Subgroups

	subgroup	G/H
center	C₂×C₇	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#137	C₂
socle	C₂×C₃×C₇	#121
prime radical	C₃×C₇×#41	C₂²
hypercenter	C₇×#137	D₃
hypocenter	C₃	C₇×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#137

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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)(11,14,17,31)(12,19,22,37)(13,24,26,40)(15,30,16,29)(18,33,34,42)(20,36,21,27)(23,39,38,35)(25,41,32,28);(11,13)(12,18)(14,28)(15,25)(16,32)(17,26)(19,35)(20,23)(21,38)(22,34)(24,30)(27,42)(29,40)(31,41)(33,36)(37,39);(9,10)(11,12,17,22)(13,23,26,38)(14,27,31,36)(15,21,16,20)(18,32,34,25)(19,30,37,29)(24,33,40,42)(28,39,41,35)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5526);