C₃×Dic₃₇₇ #17985295

order	4524
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₃₇₇

Splittings

derived, Hall	C₁₃×C₂₉ ⋊ C₃×C₄
Hall	C₃×C₁₃×C₂₉ ⋊ C₄
Hall	C₃×C₂₉ ⋊ Dic₁₃
Hall	C₃×C₁₃ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ C₃×Dic₁₃
Sylow	C₁₃ ⋊ C₃×Dic₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃₇₇
commutator, hypocenter	C₁₃×C₂₉	C₃×C₄
Frattini	C₂	C₃×D₃₇₇
Fitting, socle, prime radical	C₂×C₃×C₁₃×C₂₉	C₂
cosocle	C₂×C₁₃×C₂₉	C₂×C₃
LCS	G ▷ C₁₃×C₂₉
UCS	C₁ ◁ C₂×C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 4524

A minimal faithful permutation representation — 2 generators on 49 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);(4,6,5,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

1140 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×Dic₃₇₇, Dic₃₇₇⋊C₉, C₃×Dic₃₇₇⋊C₃, …
maximal normal subgroup of	C₉×Dic₃₇₇, Dic₃₇₇⋊C₉, C₃×Dic₃₇₇⋊C₃, …
maximal quotient of	C₉×Dic₃₇₇, C₃²×Dic₃₇₇, C₃×Dic₁₁₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4524, 5);

C3×Dic377 #17985295