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C₇×C₁₉×C₂²⋊C₉ #18012137

order	4788
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₇×C₉×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇×C₁₉	A₄
commutator, hypocenter	C₂²	C₇×C₉×C₁₉
Frattini	C₃	C₇×A₄×C₁₉
Fitting, socle, prime radical	C₂²×C₃×C₇×C₁₉	C₃
cosocle	C₂²×C₃	C₃×C₇×C₁₉
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₇×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)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(28,29,30)(31,32,34,33,35,37,36,38,39);(27,30)(28,29)

Conjugacy classes & character table

1596 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×C₇×C₁₉×C₂²⋊C₉, C₁₉×C₂²⋊C₉×C₄₉, C₇×C₁₉×C₂²⋊(C₇⋊C₉), …
maximal normal subgroup of	C₅×C₇×C₁₉×C₂²⋊C₉, C₁₉×C₂²⋊C₉×C₄₉, C₇²×C₁₉×C₂²⋊C₉
maximal quotient of	C₅×C₇×C₁₉×C₂²⋊C₉, C₁₉×C₂²⋊C₉×C₄₉, C₇×C₁₉×C₂²⋊(C₇⋊C₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 53);