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C₃×C₉×D₇ #67376

order	378
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

C₃×C₉×D₇

Splittings

Fitting, Hall	C₃×C₇×C₉ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₉	D₇
commutator, hypocenter	C₇	C₂×C₃×C₉
Frattini	C₃	C₃²×D₇
Fitting	C₃×C₇×C₉	C₂
socle	C₃²×C₇	C₂×C₃
prime radical	C₃²×D₇	C₃
cosocle	C₃×C₇	C₂×C₃²
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 2 generators on 19 points:

(5,6)(7,8)(9,10)(11,12,14,13,15,17,16,18,19);(1,2,3)(4,5,7,9,10,8,6)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₇×D₉, D₃×C₉×D₇, D₇×C₃⋊D₉, …
maximal normal subgroup of	C₃×D₇×D₉, D₃×C₉×D₇, D₇×C₃⋊D₉, …
maximal quotient of	C₃×C₉×Dic₇, C₂×C₃×C₉×D₇, C₉²×D₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 29);