C₇×#4908 #13055943

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

Aliases

direct product

C₇×#4908

Splittings

Fitting	C₃×C₇×M₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #509
Sylow	C₃ ⋊ C₇×#509

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×D₂₄
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×M₃₂

180 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5232);