C₂×D₇₆₃ #15690239

order	3052
automorphism	Aut(G) order 988848
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₅₂₆
direct product	C₂×D₇₆₃
unambiguous semidirect product	(C₂×C₇)⋊D₁₀₉
unambiguous semidirect product	(C₂×C₁₀₉)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₁₀₉ ⋊ C₂
derived, Hall	C₇×C₁₀₉ ⋊ C₂²
Sylow	C₁₀₉ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₆₃
commutator, cosocle, hypocenter	C₇×C₁₀₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₀₉	C₂
LCS	G ▷ C₇×C₁₀₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 1526

A minimal faithful permutation representation — 2 generators on 118 points:

(1,2)(3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(4,9)(5,8)(6,7)(11,118)(12,117)(13,116)(14,115)(15,114)(16,113)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₅×D₇₆₃, C₂×D₃₈₁₅, C₂×C₅×D₇₆₃, …
maximal normal subgroup of	C₂×C₅×D₇₆₃, C₂×C₇×D₇₆₃, C₂×C₁₁×D₇₆₃, …
maximal quotient of	D₅×D₇₆₃, C₂×D₃₈₁₅, C₂×C₅×D₇₆₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3052, 9);

C2×D763 #15690239