C₁₉×D₃₃₅ #27163447

order	12730
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₃₃₅
unambiguous semidirect product	C₅⋊(C₁₉×D₆₇)
unambiguous semidirect product	C₆₇⋊(D₅×C₁₉)

Splittings

Fitting, socle, Hall	C₅×C₁₉×C₆₇ ⋊ C₂
derived, Hall	C₅×C₆₇ ⋊ C₂×C₁₉
Hall	C₁₉×C₆₇ ⋊ D₅
Hall	C₅×C₁₉ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₅×C₁₉
Sylow	C₅ ⋊ C₁₉×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₃₃₅
commutator, cosocle, hypocenter	C₅×C₆₇	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₉×C₆₇	C₂
LCS	G ▷ C₅×C₆₇
UCS	C₁ ◁ C₁₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 6365

A minimal faithful permutation representation — 2 generators on 91 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)(69,70)(71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(68,69,71,72,70)

Conjugacy classes & character table

3211 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅×C₁₉×D₆₇, D₁₉×D₃₃₅, D₉₅⋊D₆₇, …
maximal normal subgroup of	D₅×C₁₉×D₆₇, D₁₉×D₃₃₅, D₉₅⋊D₆₇, …
maximal quotient of	C₁₉×Dic₃₃₅, C₂×C₁₉×D₃₃₅, C₃×C₁₉×D₃₃₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12730, 6);

C19×D335 #27163447