C₇×#4651 #13055688

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#4651

Splittings

Hall	C₃×C₇ ⋊ #383
Sylow	C₃ ⋊ C₇×#383

Subgroups

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#95	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₇×#95	D₃
hypocenter	C₃	C₇×#383
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#95

276 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 126 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)(16,20)(19,22)(25,29)(27,30);(15,17,18,21)(16,19,20,22)(23,25,26,29)(24,27,28,30);(9,10)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,27,29,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4977);