D₃×C₇×C₇₉ #15693292

order	3318
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₇₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₇×C₇₉
unambiguous semidirect product	C₃⋊(C₂×C₇×C₇₉)

Splittings

Fitting, socle, Hall	C₃×C₇×C₇₉ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₇×C₇₉
Hall	C₃×C₇₉ ⋊ C₂×C₇
Hall	C₃×C₇ ⋊ C₂×C₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₇₉	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₇×C₇₉
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₉	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₇×C₇₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 1659

A minimal faithful permutation representation — 2 generators on 89 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);(2,3)

Conjugacy classes & character table

1659 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₃×C₇×C₇₉, C₇×D₁₅×C₇₉, D₃×C₄₉×C₇₉, …
maximal normal subgroup of	C₅×D₃×C₇×C₇₉, D₃×C₄₉×C₇₉, D₃×C₇²×C₇₉, …
maximal quotient of	C₅×D₃×C₇×C₇₉, C₇×D₁₅×C₇₉, D₃×C₄₉×C₇₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 9);

D3×C7×C79 #15693292