C₃×#89303 #13051167

order	1344
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89303

Splittings

Fitting	C₂×C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #350
Sylow	C₇ ⋊ C₃×#350

Subgroups

	subgroup	G/H
center	C₃×C₈	D₂₈
commutator	C₄×C₇	C₂×C₃×C₈
Frattini	C₂×C₈	C₂×C₃×D₇
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	C₂²⋊C₈
prime radical	C₃×#1157	C₄
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	C₂×C₃×C₁₆	D₇
hypocenter	C₇	C₃×#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 456);