C₇×#5239 #13056271

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

Aliases

direct product

C₇×#5239

Splittings

Hall	C₃×C₇ ⋊ #452
Sylow	C₃ ⋊ C₇×#452

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#41

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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)(15,18,20,19)(16,22,23,27)(17,25,26,28)(21,29,30,24);(15,17)(16,21)(18,25)(19,28)(20,26)(22,29)(23,30)(24,27);(9,10)(11,12)(13,14)(15,16,20,23)(17,24,26,29)(18,22,19,27)(21,28,30,25)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5560);