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C₃×#6374 #10620557

order	672
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ SD₃₂
Sylow	C₇ ⋊ C₃×SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₃	D₅₆
commutator, cosocle	C₇×C₈	C₂²×C₃
Frattini	C₈	C₂×C₃×D₇
Fitting	C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	D₈
prime radical	C₃×D₅₆	C₂
hypercenter	C₃×C₁₆	D₇
hypocenter	C₇	C₃×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×C₁₆

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6374, #13052198, #13052201, …
maximal normal subgroup of	D₃×#6374, #13052198, #13052201, …
maximal quotient of	C₃×#89300, C₃×#89304, C₃×#89298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 139);