C₂×D₃₂₃ #13046277

order	1292
isoclinism	isoclinic class of D₃₂₃
automorphism	Aut(G) order 186048
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₄₆
direct product	C₂×D₃₂₃
unambiguous semidirect product	(C₂×C₁₇)⋊D₁₉
unambiguous semidirect product	(C₂×C₁₉)⋊D₁₇

Splittings

Fitting, socle	C₂×C₁₇×C₁₉ ⋊ C₂
derived, Hall	C₁₇×C₁₉ ⋊ C₂²
Sylow	C₁₉ ⋊ C₂×D₁₇
Sylow	C₁₇ ⋊ C₂×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₃
commutator, cosocle, hypocenter	C₁₇×C₁₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₇×C₁₉	C₂
LCS	G ▷ C₁₇×C₁₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 646

A minimal faithful permutation representation — 2 generators on 38 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);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₇⋊(D₁₉⋊C₃), D₃×D₃₂₃, C₂×C₃×D₃₂₃, …
maximal normal subgroup of	C₂×C₁₇⋊(D₁₉⋊C₃), C₂×C₃×D₃₂₃, C₂×C₁₁×D₃₂₃, …
maximal quotient of	D₃×D₃₂₃, C₂×C₃×D₃₂₃, C₂×D₉₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1292, 10);

C2×D323 #13046277