C₂×D₈₃₃ #16809708

order	3332
automorphism	Aut(G) order 1119552
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₆₆₆
direct product	C₂×D₈₃₃
unambiguous semidirect product	(C₂×C₁₇)⋊D₄₉
unambiguous semidirect product	(C₂×C₄₉)⋊D₁₇

Splittings

Fitting	C₂×C₁₇×C₄₉ ⋊ C₂
derived, Hall	C₁₇×C₄₉ ⋊ C₂²
Sylow	C₄₉ ⋊ C₂×D₁₇
Sylow	C₁₇ ⋊ C₂×D₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₃₃
commutator, cosocle, hypocenter	C₁₇×C₄₉	C₂²
Frattini	C₇	C₂×D₁₁₉
Fitting	C₂×C₁₇×C₄₉	C₂
socle	C₂×C₇×C₁₇	D₇
LCS	G ▷ C₁₇×C₄₉
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 1666

A minimal faithful permutation representation — 2 generators on 68 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);(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₅×D₈₃₃, C₂×D₄₁₆₅, C₂×C₅×D₈₃₃, …
maximal normal subgroup of	C₂×C₅×D₈₃₃, C₂×C₁₁×D₈₃₃, D₃₃₃₂
maximal quotient of	D₅×D₈₃₃, C₂×D₄₁₆₅, C₂×C₅×D₈₃₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3332, 11);

C2×D833 #16809708