C₃×#89462 #13055123

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#89462

Splittings

Fitting	C₃×C₄×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #433
Sylow	C₇ ⋊ C₃×#433

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂²×C₃×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×C₄×C₈	D₇
hypocenter	C₇	C₃×#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈

192 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 34
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 34 points:

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

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, 4412);