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C₄×C₁₉×ΓL(1,𝔽₈) #13097591

order	1596
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×C₁₉×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₄×C₇)⋊(C₃×C₁₉)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃×C₄×C₁₉
Frattini	C₂	C₂×C₁₉×ΓL(1,𝔽₈)
Fitting	C₄×C₇×C₁₉	C₃
socle	C₂×C₇×C₁₉	C₂×C₃
prime radical	C₂×C₁₉×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇	C₂×C₃×C₁₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₄×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 532

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×ΓL(1,𝔽₈)×C₁₉⋊C₃, Dic₃×C₁₉×ΓL(1,𝔽₈), C₃×C₄×C₁₉×ΓL(1,𝔽₈), …
maximal normal subgroup of	C₄×ΓL(1,𝔽₈)×C₁₉⋊C₃, C₃×C₄×C₁₉×ΓL(1,𝔽₈), C₄×C₇×C₁₉×ΓL(1,𝔽₈), …
maximal quotient of	C₄×C₁₉×C₇⋊C₉, Dic₃×C₁₉×ΓL(1,𝔽₈), C₃×C₄×C₁₉×ΓL(1,𝔽₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1596, 14);