C₃×#89292 #13051157

order	1344
isoclinism	isoclinic class of #6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89292

Splittings

Fitting	C₃×C₇×#108 ⋊ C₂
Hall	C₃×C₇ ⋊ #359
Sylow	C₇ ⋊ C₃×#359

Subgroups

	subgroup	G/H
center	C₃×C₄	#1141
commutator	C₇×C₈	C₂×C₃×C₄
Frattini	C₂×C₈	C₂×C₃×D₇
Fitting	C₃×C₇×#108	C₂
socle	C₂×C₃×C₇	#102
prime radical	C₃×#6467	C₂
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	C₃×#108	D₇
hypocenter	C₇	C₃×#359
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#108

112 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 50 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)(4,5,7,9,10,8,6)(11,13,14,23,17,25,16,32)(12,18,19,33,21,35,31,34)(15,28,26,40,30,42,29,24)(20,36,27,22,37,38,41,39);(5,6)(7,8)(9,10)(11,12)(13,22)(14,19)(15,27)(16,31)(17,21)(18,24)(20,29)(23,38)(25,39)(26,37)(28,33)(30,41)(32,36)(34,42)(35,40)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 446);