C₇×#5086 #13056120

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#5086

Splittings

Fitting	C₃×C₇×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #497
Sylow	C₃ ⋊ C₇×#497

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×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×D₁₂	C₂²
hypercenter	C₇×C₄⋊Q₈	D₃
hypocenter	C₃	C₇×#497
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₄⋊Q₈

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

normal subgroups only

This group has 248 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)(15,17,19,25)(16,21,20,29)(18,27,28,24)(22,30,26,23);(11,12,13,14)(16,20)(21,29)(22,26)(23,30);(9,10)(12,14)(15,16)(17,23)(18,26)(19,20)(21,24)(22,28)(25,30)(27,29)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5409);