C₇×#4561 #13051343

order	1344
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#4561

Splittings

Hall	C₃×C₇ ⋊ #350
Sylow	C₃ ⋊ C₇×#350

Subgroups

	subgroup	G/H
center	C₇×C₈	#67
commutator	C₃×C₄	C₂×C₇×C₈
Frattini	C₂×C₈	C₂×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	C₂²⋊C₈
prime radical	C₃×C₇×#41	C₄
cosocle	C₂×C₃×C₈	C₂²×C₇
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#350
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₇×#131

112 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(11,13)(12,18)(14,23)(15,30)(16,24)(17,25)(19,32)(20,33)(21,34)(22,29)(26,40)(27,31)(28,38)(35,41)(36,42)(37,39);(9,10)(11,12,14,19,16,20,27,35,17,21,28,36,15,29,26,39)(13,22,23,37,24,18,31,32,25,33,38,41,30,34,40,42)

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, 632);