C₇×#5230 #13056264

order	1344
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5230

Splittings

Hall	C₃×C₇ ⋊ #395
Sylow	C₃ ⋊ C₇×#395

Subgroups

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂³×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₂²×C₇×Q₈	D₃
hypocenter	C₃	C₇×#395
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₇×Q₈

364 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 1344

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5553);