C₂×D₅₆₇ #72210122

order	2268
automorphism	Aut(G) order 367416
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₃₄
direct product	C₂×D₅₆₇

Splittings

Fitting	C₂×C₇×C₈₁ ⋊ C₂
derived, Hall	C₇×C₈₁ ⋊ C₂²
Sylow	C₈₁ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₆₇
commutator, cosocle, hypocenter	C₇×C₈₁	C₂²
Frattini	C₂₇	C₂×D₂₁
Fitting	C₂×C₇×C₈₁	C₂
socle	C₂×C₃×C₇	D₂₇
LCS	G ▷ C₇×C₈₁
UCS	C₁ ◁ C₂

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 1134

A minimal faithful permutation representation — 2 generators on 90 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);(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,90)(86,89)(87,88)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₂₂₆₈, C₂×D₁₇₀₁, C₂×D₂₈₃₅, …
maximal normal subgroup of	D₂₂₆₈
maximal quotient of	D₂₂₆₈, C₂×D₁₇₀₁, Dic₁₁₃₄, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,90)(86,89)(87,88));

C2×D567 #72210122