C₃×Dic₁₀₆ #11929666

order	1272
isoclinism	isoclinic class of Dic₁₀₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×Dic₁₀₆
unambiguous semidirect product	C₅₃⋊(C₃×Q₈)

Splittings

Hall	C₃×C₅₃ ⋊ Q₈
Sylow	C₅₃ ⋊ C₃×Q₈

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, prime radical	C₂×C₃×C₅₃	C₂²
hypercenter	C₃×C₄	D₅₃
hypocenter	C₅₃	C₃×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1272

A minimal faithful permutation representation — 2 generators on 64 points:

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

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, 15);

C3×Dic106 #11929666