C₇×#5047 #13056086

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

Aliases

direct product

C₇×#5047

Splittings

Fitting	C₃×C₇×C₂²⋊Q₈ ⋊ C₂
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	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×#235	C₂
hypercenter	C₇×C₂²⋊Q₈	D₃
hypocenter	C₃	C₇×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×C₂²⋊Q₈

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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)(15,18,20,26)(16,22,23,30)(17,24,19,28)(21,29,27,25);(11,12)(13,14)(15,17,20,19)(16,21,23,27)(18,24,26,28)(22,29,30,25);(9,10)(15,16)(17,21)(18,25)(19,27)(20,23)(22,24)(26,29)(28,30)

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, 5375);