C₂×D₃₂₅ #13049912

order	1300
isoclinism	isoclinic class of D₃₂₅
automorphism	Aut(G) order 156000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₅₀
direct product	C₂×D₃₂₅
unambiguous semidirect product	(C₂×C₁₃)⋊D₂₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₁₃

Splittings

Fitting	C₂×C₁₃×C₂₅ ⋊ C₂
derived, Hall	C₁₃×C₂₅ ⋊ C₂²
Sylow	C₂₅ ⋊ C₂×D₁₃
Sylow	C₁₃ ⋊ C₂×D₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₅
commutator, cosocle, hypocenter	C₁₃×C₂₅	C₂²
Frattini	C₅	C₂×D₆₅
Fitting	C₂×C₁₃×C₂₅	C₂
socle	C₂×C₅×C₁₃	D₅
LCS	G ▷ C₁₃×C₂₅
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 650

A minimal faithful permutation representation — 2 generators on 40 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);(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₂₅⋊(D₁₃⋊C₃), D₃×D₃₂₅, C₂×C₃×D₃₂₅, …
maximal normal subgroup of	C₂×C₂₅⋊(D₁₃⋊C₃), C₂×C₃×D₃₂₅, C₂×C₁₁×D₃₂₅, …
maximal quotient of	D₃×D₃₂₅, C₂×C₃×D₃₂₅, C₂×D₉₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1300, 14);

C2×D325 #13049912