C₇×#5234 #13056272

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5234

Splittings

Hall	C₃×C₇ ⋊ #408
Sylow	C₃ ⋊ C₇×#408

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#408
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41

316 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5561);