Dic₄₇₁₅ #34038195

order	18860
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₇₁₅
unambiguous semidirect product	C₂₃⋊Dic₂₀₅

Splittings

derived, Hall	C₅×C₂₃×C₄₁ ⋊ C₄
Hall	C₂₃×C₄₁ ⋊ Dic₅
Hall	C₅×C₄₁ ⋊ Dic₂₃
Hall	C₅×C₂₃ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₁₅
Sylow	C₂₃ ⋊ Dic₂₀₅
Sylow	C₅ ⋊ Dic₉₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₇₁₅
commutator, hypocenter	C₅×C₂₃×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₃×C₄₁	C₂
LCS	G ▷ C₅×C₂₃×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 18860

A minimal faithful permutation representation — 2 generators on 73 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,3,2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,5,23,41,2|a^2=e,b^a=b^4,c^a=c^22,d^a=d^40

Conjugacy classes & character table

4718 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₄₃₀, Dic₁₄₁₄₅
maximal normal subgroup of	Dic₉₄₃₀
maximal quotient of	Dic₁₄₁₄₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18860, 9);

Dic4715 #34038195