C₇×D₁₀₁ #13083451

order	1414
isoclinism	isoclinic class of D₁₀₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₀₁
unambiguous semidirect product	C₁₀₁⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇×C₁₀₁ ⋊ C₂
derived, Sylow	C₁₀₁ ⋊ C₂×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₀₁
commutator, cosocle, hypocenter	C₁₀₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₀₁	C₂
LCS	G ▷ C₁₀₁
UCS	C₁ ◁ C₇

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 707

A minimal faithful permutation representation — 2 generators on 108 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108);(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×#87938, C₇⋊#87938, D₇×D₁₀₁, …
maximal normal subgroup of	C₇×#87938, C₇⋊#87938, D₇×D₁₀₁, …
maximal quotient of	C₇×Dic₁₀₁, C₂×C₇×D₁₀₁, C₃×C₇×D₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1414, 2);

C7×D101 #13083451