C₇×#5082 #13056117

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

Aliases

direct product

C₇×#5082

Splittings

Hall	C₃×C₇ ⋊ #491
Sylow	C₃ ⋊ C₇×#491

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, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₇×C₄⋊Q₈	D₃
hypocenter	C₃	C₇×#491
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₄⋊Q₈

180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 34
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5,6,7)(8,9,10)(12,16)(13,17)(19,21,23,29)(20,25,24,32)(22,30,31,28)(26,34,33,27);(11,13,14,17)(12,15,16,18)(20,24)(25,32)(26,33)(27,34);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,22,26,23,24,31,33)(21,27,28,32,29,34,30,25)

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, 5406);