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C₃²×C₁₉×Dic₇ #18012199

order	4788
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₁₉ · Schur multiplier (H₂) = C₃

Aliases

C₃²×C₁₉×Dic₇

Splittings

derived, Sylow	C₇ ⋊ C₃²×C₄×C₁₉
Hall	C₃²×C₇×C₁₉ ⋊ C₄
Hall	C₇×C₁₉ ⋊ C₃²×C₄
Hall	C₃²×C₇ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₁₉	D₇
commutator, hypocenter	C₇	C₃²×C₄×C₁₉
Frattini	C₂	C₃²×D₇×C₁₉
Fitting, socle, prime radical	C₂×C₃²×C₇×C₁₉	C₂
cosocle	C₂×C₇	C₂×C₃²×C₁₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃²×C₁₉

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 4788

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

1710 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×C₁₉×Dic₃₅, C₃²×C₅×C₁₉×Dic₇, C₃²×C₁₉×C₇⋊AGL(1,𝔽₅), …
maximal normal subgroup of	C₃²×C₅×C₁₉×Dic₇, C₃²×C₇×C₁₉×Dic₇
maximal quotient of	C₃²×C₁₉×Dic₃₅, C₃²×C₅×C₁₉×Dic₇, C₃²×C₁₉×C₇⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 115);