C₃×D₂₁₂ #11929668

order	1272
isoclinism	isoclinic class of Dic₁₀₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×D₂₁₂
unambiguous semidirect product	C₄⋊(C₃×D₅₃)

Splittings

Fitting	C₃×C₄×C₅₃ ⋊ C₂
Hall	C₃×C₅₃ ⋊ D₄
Sylow	C₅₃ ⋊ C₃×D₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₅₃
commutator, cosocle	C₂×C₅₃	C₂²×C₃
Frattini	C₂	C₂×C₃×D₅₃
Fitting	C₃×C₄×C₅₃	C₂
socle	C₂×C₃×C₅₃	C₂²
hypercenter	C₃×C₄	D₅₃
hypocenter	C₅₃	C₃×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 636

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 17);

C3×D212 #11929668