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C₂×C₂²⋊(C₇⋊C₉) #92636

order	504
isoclinism	isoclinic class of C₂²⋊ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×A₄×AGL(1,𝔽₇) (order 1512) · Inn(G) = C₂²⋊ΓL(1,𝔽₈) · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₂²⋊(C₇⋊C₉)
unambiguous semidirect product	(C₂×C₇)⋊(C₂²⋊C₉)

Splittings

derived	C₂²×C₇ ⋊ C₂×C₉
Hall	C₂³×C₇ ⋊ C₉
Sylow	C₂³ ⋊ C₇⋊C₉
Sylow	C₇ ⋊ C₂×C₂²⋊C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₂×C₉
Frattini	C₃	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting, socle, prime radical	C₂³×C₃×C₇	C₃
cosocle	C₂²×C₃×C₇	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂×C₃

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₇⋊(C₂²⋊C₉), C₄×C₂²⋊(C₇⋊C₉), C₂×D₇⋊(C₂²⋊C₉), …
maximal normal subgroup of	Dic₇⋊(C₂²⋊C₉), C₄×C₂²⋊(C₇⋊C₉), C₂×D₇⋊(C₂²⋊C₉), …
maximal quotient of	C₄×C₂²⋊(C₇⋊C₉), C₂×C₇⋊(Q₈⋊C₉), #41⋊(C₇⋊C₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 62);