Dic₃₃₁₅ #27210467

order	13260
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₁₅
unambiguous semidirect product	C₃⋊Dic₁₁₀₅

Splittings

derived, Hall	C₃×C₅×C₁₃×C₁₇ ⋊ C₄
Hall	C₅×C₁₃×C₁₇ ⋊ Dic₃
Hall	C₃×C₁₃×C₁₇ ⋊ Dic₅
Hall	C₃×C₅×C₁₇ ⋊ Dic₁₃
Hall	C₁₃×C₁₇ ⋊ Dic₁₅
Hall	C₃×C₅×C₁₃ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₃₉
Hall	C₅×C₁₃ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₆₅
Hall	C₃×C₁₃ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₁₉₅
Hall	C₃×C₅ ⋊ Dic₂₂₁
Sylow	C₁₃ ⋊ Dic₂₅₅
Sylow	C₅ ⋊ Dic₆₆₃
Sylow	C₃ ⋊ Dic₁₁₀₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃₁₅
commutator, hypocenter	C₃×C₅×C₁₃×C₁₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₃×C₁₇	C₂
LCS	G ▷ C₃×C₅×C₁₃×C₁₇
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 13260

A minimal faithful permutation representation — 2 generators on 42 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2,3,5,13,17,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^12,e^a=e^16

Conjugacy classes & character table

3318 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₆₃₀
maximal normal subgroup of	Dic₆₆₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13260, 17);

Dic3315 #27210467