C₇×D₂₃₇ #15693293

order	3318
isoclinism	isoclinic class of D₂₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₂₃₇
unambiguous semidirect product	C₃⋊(C₇×D₇₉)
unambiguous semidirect product	C₇₉⋊(D₃×C₇)

Splittings

Fitting, socle, Hall	C₃×C₇×C₇₉ ⋊ C₂
derived, Hall	C₃×C₇₉ ⋊ C₂×C₇
Hall	C₇×C₇₉ ⋊ D₃
Sylow	C₇₉ ⋊ D₃×C₇
Hall	C₃×C₇ ⋊ D₇₉
Sylow	C₃ ⋊ C₇×D₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₂₃₇
commutator, cosocle, hypocenter	C₃×C₇₉	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₉	C₂
LCS	G ▷ C₃×C₇₉
UCS	C₁ ◁ C₇

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₇×D₂₃₇, C₇×D₁₁₈₅, C₄₉×D₂₃₇, …
maximal normal subgroup of	C₅×C₇×D₂₃₇, C₄₉×D₂₃₇, C₇²×D₂₃₇, …
maximal quotient of	C₅×C₇×D₂₃₇, C₇×D₁₁₈₅, C₄₉×D₂₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 10);

C7×D237 #15693293