C₂₉×D₅₇ #15693254

order	3306
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₉×D₅₇
unambiguous semidirect product	C₃⋊(C₂₉×D₁₉)
unambiguous semidirect product	C₁₉⋊(D₃×C₂₉)

Splittings

Fitting, socle, Hall	C₃×C₁₉×C₂₉ ⋊ C₂
derived, Hall	C₃×C₁₉ ⋊ C₂×C₂₉
Hall	C₁₉×C₂₉ ⋊ D₃
Hall	C₃×C₂₉ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₃×C₂₉
Sylow	C₃ ⋊ C₂₉×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂₉	D₅₇
commutator, cosocle, hypocenter	C₃×C₁₉	C₂×C₂₉
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₂₉	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₂₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1653

A minimal faithful permutation representation — 2 generators on 51 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);(31,32)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₂₉×D₅₇, C₂₉×D₂₈₅, C₇×C₂₉×D₅₇, …
maximal normal subgroup of	C₅×C₂₉×D₅₇, C₇×C₂₉×D₅₇, D₅₇×C₂₉⋊C₇, …
maximal quotient of	C₅×C₂₉×D₅₇, C₂₉×D₂₈₅, C₇×C₂₉×D₅₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3306, 8);

C29×D57 #15693254