C₇×#5002 #13056037

order	1344
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5002

Splittings

Fitting	C₃×C₇×#117 ⋊ C₂
Hall	C₃×C₇ ⋊ #416
Sylow	C₃ ⋊ C₇×#416

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#67
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#416
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#117

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

normal subgroups only

This group has 316 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) = 672

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

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

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, 5326);