C₃×C₁₉×Dic₅₉ #27212092

order	13452
isoclinism	isoclinic class of D₅₉
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₁₉×Dic₅₉
unambiguous semidirect product	C₅₉⋊(C₃×C₄×C₁₉)

Splittings

derived, Sylow	C₅₉ ⋊ C₃×C₄×C₁₉
Hall	C₃×C₁₉×C₅₉ ⋊ C₄
Hall	C₁₉×C₅₉ ⋊ C₃×C₄
Hall	C₃×C₅₉ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₉	D₅₉
commutator, hypocenter	C₅₉	C₃×C₄×C₁₉
Frattini	C₂	C₃×C₁₉×D₅₉
Fitting, socle, prime radical	C₂×C₃×C₁₉×C₅₉	C₂
cosocle	C₂×C₅₉	C₂×C₃×C₁₉
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₂×C₃×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13452

A minimal faithful permutation representation — 2 generators on 85 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)(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);(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)

Conjugacy classes & character table

3534 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 3);

C3×C19×Dic59 #27212092