C₇×#4985 #13056016

order	1344
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4985

Splittings

Hall	C₃×C₇ ⋊ #400
Sylow	C₃ ⋊ C₇×#400

Subgroups

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂³×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₇×C₄⋊C₄	D₃
hypocenter	C₃	C₇×#400
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₇×C₄⋊C₄

268 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

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

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, 5305);