C₂×D₃₃₃ #13050623

order	1332
isoclinism	isoclinic class of D₃₃₃
automorphism	Aut(G) order 143856
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₆₆
direct product	C₂×D₃₃₃
unambiguous semidirect product	(C₂×C₉)⋊D₃₇
unambiguous semidirect product	(C₂×C₃₇)⋊D₉

Splittings

Fitting	C₂×C₉×C₃₇ ⋊ C₂
derived, Hall	C₉×C₃₇ ⋊ C₂²
Sylow	C₃₇ ⋊ C₂×D₉
Sylow	C₉ ⋊ C₂×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₃₃
commutator, cosocle, hypocenter	C₉×C₃₇	C₂²
Frattini	C₃	C₂×D₁₁₁
Fitting	C₂×C₉×C₃₇	C₂
socle	C₂×C₃×C₃₇	D₃
LCS	G ▷ C₉×C₃₇
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 666

A minimal faithful permutation representation — 2 generators on 48 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);(4,11)(5,10)(6,9)(7,8)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₁×D₃₃₃, C₂×C₁₁×D₃₃₃, C₂×D₃₆₆₃, …
maximal normal subgroup of	C₂×C₁₁×D₃₃₃, C₂×C₁₃×D₃₃₃, C₂×C₁₇×D₃₃₃, …
maximal quotient of	D₁₁×D₃₃₃, C₂×C₁₁×D₃₃₃, C₂×D₃₆₆₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1332, 17);

C2×D333 #13050623