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C₃×D₅₆ #66592

order	336
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₃×D₅₆

Splittings

Fitting	C₃×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ D₈
Sylow	C₇ ⋊ C₃×D₈

Subgroups

	subgroup	G/H
center	C₂×C₃	D₂₈
commutator, cosocle	C₄×C₇	C₂²×C₃
Frattini	C₄	C₂×C₃×D₇
Fitting	C₃×C₇×C₈	C₂
socle	C₂×C₃×C₇	D₄
hypercenter	C₃×C₈	D₇
hypocenter	C₇	C₃×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3)(4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18);(5,11)(6,10)(7,9)(13,18)(14,17)(15,16)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620472, #10620473, #10620475, …
maximal normal subgroup of	#10620472, #10620473, #10620475, …
maximal quotient of	C₃×D₁₁₂, C₃×#6374, C₃×Dic₅₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 61);