C₂×D₄₉₇ #14521908

order	1988
isoclinism	isoclinic class of D₄₉₇
automorphism	Aut(G) order 417480
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₉₉₄
direct product	C₂×D₄₉₇
unambiguous semidirect product	(C₂×C₇)⋊D₇₁
unambiguous semidirect product	(C₂×C₇₁)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₇₁ ⋊ C₂
derived, Hall	C₇×C₇₁ ⋊ C₂²
Sylow	C₇₁ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₉₇
commutator, cosocle, hypocenter	C₇×C₇₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₇₁	C₂
LCS	G ▷ C₇×C₇₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 994

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₃₄₇₉, C₂×C₇⋊(D₇₁⋊C₇), D₇×D₄₉₇, …
maximal normal subgroup of	C₂×C₇⋊(D₇₁⋊C₇), C₂×C₇×D₄₉₇, C₂×C₁₁×D₄₉₇, …
maximal quotient of	C₂×D₃₄₇₉, D₇×D₄₉₇, C₂×C₇⋊D₄₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1988, 12);

C2×D497 #14521908