C₃×D₇₆₇ #17987175

order	4602
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₇₆₇
unambiguous semidirect product	C₅₉⋊(C₃×D₁₃)

Splittings

Fitting, socle, Hall	C₃×C₁₃×C₅₉ ⋊ C₂
derived, Hall	C₁₃×C₅₉ ⋊ C₂×C₃
Hall	C₃×C₅₉ ⋊ D₁₃
Sylow	C₅₉ ⋊ C₃×D₁₃
Hall	C₃×C₁₃ ⋊ D₅₉
Sylow	C₁₃ ⋊ C₃×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₇₆₇
commutator, cosocle, hypocenter	C₁₃×C₅₉	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₅₉	C₂
LCS	G ▷ C₁₃×C₅₉
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 2301

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Conjugacy classes & character table

1155 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×D₇₆₇, C₅₉⋊(D₁₃⋊C₉), C₃×(C₁₃⋊C₃)⋊D₅₉, …
maximal normal subgroup of	C₉×D₇₆₇, C₅₉⋊(D₁₃⋊C₉), C₃×(C₁₃⋊C₃)⋊D₅₉, …
maximal quotient of	C₉×D₇₆₇, C₃×D₂₃₀₁, C₃²×D₇₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4602, 5);

C3×D767 #17987175