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C₉×D₇₇ #13063452

order	1386
isoclinism	isoclinic class of D₇₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₇₇
unambiguous semidirect product	C₁₁⋊(C₉×D₇)

Splittings

Fitting, Hall	C₇×C₉×C₁₁ ⋊ C₂
derived, Hall	C₇×C₁₁ ⋊ C₂×C₉
Hall	C₉×C₁₁ ⋊ D₇
Hall	C₇×C₉ ⋊ D₁₁
Sylow	C₁₁ ⋊ C₉×D₇
Sylow	C₇ ⋊ C₉×D₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₇₇
commutator, hypocenter	C₇×C₁₁	C₂×C₉
Frattini	C₃	C₃×D₇₇
Fitting	C₇×C₉×C₁₁	C₂
socle, cosocle	C₃×C₇×C₁₁	C₂×C₃
prime radical	C₃×D₇₇	C₃
LCS	G ▷ C₇×C₁₁
UCS	C₁ ◁ C₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 693

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×D₇×D₁₁, D₉×D₇₇, D₆₃⋊D₁₁, …
maximal normal subgroup of	C₉×D₇×D₁₁, D₉×D₇₇, D₆₃⋊D₁₁, …
maximal quotient of	C₉×Dic₇₇, C₂×C₉×D₇₇, C₉×D₈₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1386, 7);