C₇×D₂₁₃ #15688397

order	2982
isoclinism	isoclinic class of D₂₁₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₂₁₃
unambiguous semidirect product	C₃⋊(C₇×D₇₁)

Splittings

Fitting, socle, Hall	C₃×C₇×C₇₁ ⋊ C₂
derived, Hall	C₃×C₇₁ ⋊ C₂×C₇
Hall	C₇×C₇₁ ⋊ D₃
Sylow	C₇₁ ⋊ D₃×C₇
Hall	C₃×C₇ ⋊ D₇₁
Sylow	C₃ ⋊ C₇×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₂₁₃
commutator, cosocle, hypocenter	C₃×C₇₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₁	C₂
LCS	G ▷ C₃×C₇₁
UCS	C₁ ◁ C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 1491

A minimal faithful permutation representation — 2 generators on 81 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);(9,10)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₇×D₂₁₃, C₇×D₁₀₆₅, C₇×C₃⋊(D₇₁⋊C₅), …
maximal normal subgroup of	C₅×C₇×D₂₁₃, C₇×C₃⋊(D₇₁⋊C₅), C₄₉×D₂₁₃, …
maximal quotient of	C₅×C₇×D₂₁₃, C₇×D₁₀₆₅, C₄₉×D₂₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2982, 8);

C7×D213 #15688397