C₁₉×D₅₃₁ #34118151

order	20178
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₅₃₁
unambiguous semidirect product	C₉⋊(C₁₉×D₅₉)
unambiguous semidirect product	C₅₉⋊(D₉×C₁₉)

Splittings

Fitting, Hall	C₉×C₁₉×C₅₉ ⋊ C₂
derived, Hall	C₉×C₅₉ ⋊ C₂×C₁₉
Hall	C₁₉×C₅₉ ⋊ D₉
Hall	C₉×C₁₉ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₉×C₁₉
Sylow	C₉ ⋊ C₁₉×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₅₃₁
commutator, cosocle, hypocenter	C₉×C₅₉	C₂×C₁₉
Frattini	C₃	C₁₉×D₁₇₇
Fitting	C₉×C₁₉×C₅₉	C₂
socle	C₃×C₁₉×C₅₉	D₃
LCS	G ▷ C₉×C₅₉
UCS	C₁ ◁ C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 10089

A minimal faithful permutation representation — 2 generators on 87 points:

(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)(61,63)(62,66)(64,65)(67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,68,63,66,65,64,62,61,67)

Conjugacy classes & character table

5073 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20178, 7);

C19×D531 #34118151