C₂×D₁₀₃₇ #16861900

order	4148
automorphism	Aut(G) order 1991040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₀₇₄
direct product	C₂×D₁₀₃₇
unambiguous semidirect product	(C₂×C₁₇)⋊D₆₁
unambiguous semidirect product	(C₂×C₆₁)⋊D₁₇

Splittings

Fitting, socle	C₂×C₁₇×C₆₁ ⋊ C₂
derived, Hall	C₁₇×C₆₁ ⋊ C₂²
Sylow	C₆₁ ⋊ C₂×D₁₇
Sylow	C₁₇ ⋊ C₂×D₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₀₃₇
commutator, cosocle, hypocenter	C₁₇×C₆₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₇×C₆₁	C₂
LCS	G ▷ C₁₇×C₆₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 2074

A minimal faithful permutation representation — 2 generators on 80 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);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(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

1040 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₇⋊(D₆₁⋊C₃), D₃×D₁₀₃₇, C₂×D₃₁₁₁, …
maximal normal subgroup of	C₂×C₁₇⋊(D₆₁⋊C₃), C₂×C₃×D₁₀₃₇, C₂×C₁₇⋊(D₆₁⋊C₅), …
maximal quotient of	D₃×D₁₀₃₇, C₂×D₃₁₁₁, C₂×C₃×D₁₀₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4148, 13);

C2×D1037 #16861900