C₂×D₃₄₅ #13063425

order	1380
isoclinism	isoclinic class of D₃₄₅
automorphism	Aut(G) order 121440
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₉₀
direct product	C₂×D₃₄₅
unambiguous semidirect product	(C₂×C₃)⋊D₁₁₅
unambiguous semidirect product	(C₂×C₅)⋊D₆₉
unambiguous semidirect product	(C₂×C₂₃)⋊D₁₅

Splittings

Fitting, socle	C₂×C₃×C₅×C₂₃ ⋊ C₂
derived, Hall	C₃×C₅×C₂₃ ⋊ C₂²
Hall	C₅×C₂₃ ⋊ C₂×D₃
Hall	C₃×C₂₃ ⋊ C₂×D₅
Sylow	C₂₃ ⋊ C₂×D₁₅
Hall	C₃×C₅ ⋊ C₂×D₂₃
Sylow	C₅ ⋊ C₂×D₆₉
Sylow	C₃ ⋊ C₂×D₁₁₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₄₅
commutator, cosocle, hypocenter	C₃×C₅×C₂₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₂₃	C₂
LCS	G ▷ C₃×C₅×C₂₃
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 690

A minimal faithful permutation representation — 2 generators on 33 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);(4,5)(7,10)(8,9)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₁₀₃₅, D₃×D₃₄₅, C₂×C₃⋊D₃₄₅, …
maximal normal subgroup of	C₂×C₃×D₃₄₅, C₂×C₃⋊(C₅⋊AGL(1,𝔽₂₃)), C₂×C₁₁×D₃₄₅, …
maximal quotient of	C₂×D₁₀₃₅, D₃×D₃₄₅, C₂×C₃⋊D₃₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 28);

C2×D345 #13063425