C₁₉×D₉₇ #16814770

order	3686
isoclinism	isoclinic class of D₉₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₉₇
unambiguous semidirect product	C₉₇⋊(C₂×C₁₉)

Splittings

Fitting, socle, Hall	C₁₉×C₉₇ ⋊ C₂
derived, Sylow	C₉₇ ⋊ C₂×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₉₇
commutator, cosocle, hypocenter	C₉₇	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₁₉×C₉₇	C₂
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 1843

A minimal faithful permutation representation — 2 generators on 116 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);(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₁₉×D₉₇, C₁₉×D₂₉₁, C₁₉×D₉₇⋊C₃, …
maximal normal subgroup of	C₃×C₁₉×D₉₇, C₁₉×D₉₇⋊C₃, C₁₉⋊C₃×D₉₇, …
maximal quotient of	C₃×C₁₉×D₉₇, C₁₉×D₂₉₁, C₅×C₁₉×D₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3686, 2);

C19×D97 #16814770