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C₂²×C₁₉×C₇⋊C₉ #18012134

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

Aliases

C₂²×C₁₉×C₇⋊C₉

Splittings

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

Subgroups

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

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 4788

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

(1,2)(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)(31,32,34,33,35,37,36,38,39);(3,4)(24,25,26,28,27,29,30)

Conjugacy classes & character table

1140 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₅×C₁₉×C₇⋊C₉, C₂²×C₅×C₁₉×C₇⋊C₉, C₂²×C₁₉×C₄₉⋊C₉, …
maximal normal subgroup of	C₂²×C₅×C₁₉×C₇⋊C₉, C₂²×C₇×C₁₉×C₇⋊C₉
maximal quotient of	C₂×D₅×C₁₉×C₇⋊C₉, C₂²×C₅×C₁₉×C₇⋊C₉, C₂²×C₁₉×C₄₉⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 50);