C₃×#89304 #13051168

order	1344
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89304

Splittings

Fitting	C₂×C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #357
Sylow	C₇ ⋊ C₃×#357

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₅₆
commutator	C₇×C₈	C₂×C₃×C₄
Frattini	C₂×C₈	C₂×C₃×D₇
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle	C₂²×C₃×C₇	D₈
prime radical	C₂×C₃×D₅₆	C₂
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	C₂×C₃×C₁₆	D₇
hypocenter	C₇	C₃×#357
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

132 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 457);