C₇×#4515 #13051297

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

Aliases

direct product

C₇×#4515

Splittings

Fitting	C₃×C₇×M₃₂ ⋊ C₂
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×M₃₂	C₂
socle	C₂×C₃×C₇	C₂²⋊C₈
prime radical	C₇×#235	C₄
cosocle	C₂×C₃×C₈	C₂²×C₇
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#350
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₇×M₃₂

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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)(11,13,14,23,16,24,27,38,17,25,28,39,15,30,26,40)(12,18,19,31,20,32,34,41,21,33,35,42,29,22,36,37);(9,10)(11,12)(13,22)(14,19)(15,29)(16,20)(17,21)(18,24)(23,37)(25,32)(26,36)(27,34)(28,35)(30,33)(31,38)(39,41)(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, 586);