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C₇×Dic₃×C₁₉ #13097597

order	1596
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₃×C₁₉
unambiguous semidirect product	C₃⋊(C₄×C₇×C₁₉)

Splittings

derived, Sylow	C₃ ⋊ C₄×C₇×C₁₉
Hall	C₃×C₇×C₁₉ ⋊ C₄
Hall	C₃×C₁₉ ⋊ C₄×C₇
Hall	C₃×C₇ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₁₉	D₃
commutator, hypocenter	C₃	C₄×C₇×C₁₉
Frattini	C₂	D₃×C₇×C₁₉
Fitting, socle, prime radical	C₂×C₃×C₇×C₁₉	C₂
cosocle	C₂×C₃	C₂×C₇×C₁₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₇×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 1596

A minimal faithful permutation representation — 2 generators on 33 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×C₁₉×Dic₉, C₇×Dic₃×C₁₉⋊C₃, Dic₃×C₁₉×ΓL(1,𝔽₈), …
maximal normal subgroup of	C₇×Dic₃×C₁₉⋊C₃, Dic₃×C₁₉×ΓL(1,𝔽₈), Dic₃×#87704, …
maximal quotient of	C₇×C₁₉×Dic₉, C₃×C₇×Dic₃×C₁₉, C₇×C₁₉×C₃⋊Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1596, 20);