C₃×D₇×C₈₉ #16836645

order	3738
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×D₇×C₈₉

Splittings

Fitting, socle, Hall	C₃×C₇×C₈₉ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₃×C₈₉
Hall	C₇×C₈₉ ⋊ C₂×C₃
Hall	C₃×C₇ ⋊ C₂×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈₉	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₈₉
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₈₉	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₈₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 1869

A minimal faithful permutation representation — 2 generators on 99 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(5,10)(6,9)(7,8)

Conjugacy classes & character table

1335 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×D₇×C₈₉, C₈₉×D₇⋊C₉, C₃×AGL(1,𝔽₇)×C₈₉, …
maximal normal subgroup of	C₉×D₇×C₈₉, C₈₉×D₇⋊C₉, C₃×AGL(1,𝔽₇)×C₈₉, …
maximal quotient of	C₉×D₇×C₈₉, C₃×D₂₁×C₈₉, C₃²×D₇×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3738, 4);

C3×D7×C89 #16836645