Dic₃₇₀₅ #28414734

order	14820
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₇₀₅
unambiguous semidirect product	C₃⋊Dic₁₂₃₅
unambiguous semidirect product	C₁₉⋊Dic₁₉₅

Splittings

derived, Hall	C₃×C₅×C₁₃×C₁₉ ⋊ C₄
Hall	C₅×C₁₃×C₁₉ ⋊ Dic₃
Hall	C₃×C₁₃×C₁₉ ⋊ Dic₅
Hall	C₃×C₅×C₁₉ ⋊ Dic₁₃
Hall	C₁₃×C₁₉ ⋊ Dic₁₅
Hall	C₃×C₅×C₁₃ ⋊ Dic₁₉
Hall	C₅×C₁₉ ⋊ Dic₃₉
Hall	C₅×C₁₃ ⋊ Dic₅₇
Hall	C₃×C₁₉ ⋊ Dic₆₅
Hall	C₃×C₁₃ ⋊ Dic₉₅
Sylow	C₁₉ ⋊ Dic₁₉₅
Hall	C₃×C₅ ⋊ Dic₂₄₇
Sylow	C₁₃ ⋊ Dic₂₈₅
Sylow	C₅ ⋊ Dic₇₄₁
Sylow	C₃ ⋊ Dic₁₂₃₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₇₀₅
commutator, hypocenter	C₃×C₅×C₁₃×C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₃×C₁₉	C₂
LCS	G ▷ C₃×C₅×C₁₃×C₁₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 14820

A minimal faithful permutation representation — 2 generators on 44 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Polycyclic presentation:

2,3,5,13,19,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^12,e^a=e^18

Conjugacy classes & character table

3708 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₄₁₀
maximal normal subgroup of	Dic₇₄₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14820, 20);

Dic3705 #28414734