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C₁₉×C₂²⋊(C₇⋊C₉) #18012139

order	4788
isoclinism	isoclinic class of C₂²⋊ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₉×C₂²⋊(C₇⋊C₉)
unambiguous semidirect product	C₂²⋊(C₁₉×C₇⋊C₉)
unambiguous semidirect product	C₇⋊(C₁₉×C₂²⋊C₉)

Splittings

derived, Hall	C₂²×C₇ ⋊ C₉×C₁₉
Hall	C₂²×C₇×C₁₉ ⋊ C₉
Hall	C₇×C₁₉ ⋊ C₂²⋊C₉
Hall	C₂²×C₁₉ ⋊ C₇⋊C₉
Sylow	C₇ ⋊ C₁₉×C₂²⋊C₉
Sylow	C₂² ⋊ C₁₉×C₇⋊C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₉	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₉×C₁₉
Frattini	C₃	C₁₉×C₂²⋊ΓL(1,𝔽₈)
Fitting, socle, prime radical	C₂²×C₃×C₇×C₁₉	C₃
cosocle	C₂²×C₃×C₇	C₃×C₁₉
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₁₉

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 2394

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22,23)(25,26,27)(28,30,29)(31,32,34,33,35,37,36,38,39);(20,23)(21,22)(24,25,26,28,27,29,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₁₉×C₂²⋊(C₇⋊C₉), C₁₉×C₂²⋊(C₄₉⋊C₉), C₁₉×C₂²⋊#89205, …
maximal normal subgroup of	C₅×C₁₉×C₂²⋊(C₇⋊C₉), C₇×C₁₉×C₂²⋊(C₇⋊C₉)
maximal quotient of	C₅×C₁₉×C₂²⋊(C₇⋊C₉), C₁₉×C₂²⋊(C₄₉⋊C₉), C₁₉×C₂²⋊#89205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 55);