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(C₇⋊C₉)⋊D₆₇ #22554735

order	8442
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

(C₇⋊C₉)⋊D₆₇

Splittings

derived, Hall	C₇×C₆₇ ⋊ C₂×C₉
Hall	C₇⋊C₉×C₆₇ ⋊ C₂
Hall	D₄₆₉ ⋊ C₉
Sylow	C₆₇ ⋊ D₇⋊C₉
Hall	C₇⋊C₉ ⋊ D₆₇
Sylow	C₇ ⋊ C₉×D₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	ΓL(1,𝔽₈)⋊D₆₇
commutator, hypocenter	C₇×C₆₇	C₂×C₉
Fitting, socle	C₃×C₇×C₆₇	C₂×C₃
LCS	G ▷ C₇×C₆₇
UCS	C₁ ◁ C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,7,67,3|b^3=e,c^a=c^6,d^a=d^66,c^b=c^2

Conjugacy classes & character table

516 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇⋊C₉×D₆₇, C₂×(C₇⋊C₉)⋊D₆₇
maximal normal subgroup of	D₇⋊C₉×D₆₇, C₂×(C₇⋊C₉)⋊D₆₇
maximal quotient of	(C₇⋊C₉)⋊Dic₆₇, C₂×(C₇⋊C₉)⋊D₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8442, 22);