C₇×#5030 #13056065

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#5030

Splittings

Hall	C₃×C₇ ⋊ #441
Sylow	C₃ ⋊ C₇×#441

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₄×C₇×Q₈	D₃
hypocenter	C₃	C₇×#441
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×Q₈

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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)(11,12,13,14)(17,24)(20,29)(23,30)(25,27);(15,17,19,24)(16,20,22,29)(18,27,28,25)(21,30,26,23);(9,10)(15,16,19,22)(17,23,24,30)(18,26,28,21)(20,27,29,25)

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, 5354);