D₁₉×D₁₇₇ #27212111

order	13452
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₁₉×D₁₇₇
unambiguous semidirect product	(C₃×D₁₉)⋊D₅₉
unambiguous semidirect product	(D₁₉×C₅₉)⋊D₃

Splittings

Fitting, socle, derived, Hall	C₃×C₁₉×C₅₉ ⋊ C₂²
Hall	C₁₉×C₅₉ ⋊ C₂×D₃
Hall	C₃×C₅₉ ⋊ C₂×D₁₉
Sylow	C₅₉ ⋊ D₃×D₁₉
Hall	C₃×C₁₉ ⋊ C₂×D₅₉
Sylow	C₁₉ ⋊ C₂×D₁₇₇
Sylow	C₃ ⋊ D₁₉×D₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₉×C₅₉	C₂²
LCS	G ▷ C₃×C₁₉×C₅₉
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3363

A minimal faithful permutation representation — 2 generators on 81 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)(61,62)(63,64,66,68,70,72,74,76,78,80,81,79,77,75,73,71,69,67,65);(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)(60,61,62)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 22);

D19×D177 #27212111