C₃×D₂₁₇₅ #27189384

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₁₇₅
unambiguous semidirect product	C₃⋊(C₃×D₇₂₅)
unambiguous semidirect product	C₂₅⋊(C₃×D₈₇)
unambiguous semidirect product	C₂₉⋊(C₃×D₇₅)

Splittings

Fitting, Hall	C₃²×C₂₅×C₂₉ ⋊ C₂
derived	C₃×C₂₅×C₂₉ ⋊ C₂×C₃
Hall	C₂₅×C₂₉ ⋊ C₃×D₃
Hall	C₃²×C₂₉ ⋊ D₂₅
Hall	C₃²×C₂₅ ⋊ D₂₉
Sylow	C₂₉ ⋊ C₃×D₇₅
Sylow	C₂₅ ⋊ C₃×D₈₇
Sylow	C₃² ⋊ D₇₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₂₁₇₅
commutator, cosocle, hypocenter	C₃×C₂₅×C₂₉	C₂×C₃
Frattini	C₅	C₃×D₄₃₅
Fitting	C₃²×C₂₅×C₂₉	C₂
socle	C₃²×C₅×C₂₉	D₅
LCS	G ▷ C₃×C₂₅×C₂₉
UCS	C₁ ◁ C₃

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 4350

A minimal faithful permutation representation — 2 generators on 60 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)(31,33)(32,36)(34,39)(35,42)(37,45)(38,44)(40,41)(43,49)(46,50)(47,51)(48,53)(52,54)(55,56,57)(59,60);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,50,41,53,38,36,43,34,52,33,45,35,51,47,42,37,31,54,39,49,32,44,48,40,46)(58,59,60)

Conjugacy classes & character table

3267 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₂₉×D₇₅, C₃×D₂₅×D₈₇, C₃×D₃×D₇₂₅, …
maximal normal subgroup of	C₃×D₂₉×D₇₅, C₃×D₂₅×D₈₇, C₃×D₃×D₇₂₅, …
maximal quotient of	C₃×Dic₂₁₇₅, C₂×C₃×D₂₁₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 16);

C3×D2175 #27189384