Dic₂₇₄₅ #24926668

order	10980
automorphism	Aut(G) order 7905600
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₄₅
unambiguous semidirect product	C₉⋊Dic₃₀₅

Splittings

derived, Hall	C₅×C₉×C₆₁ ⋊ C₄
Hall	C₉×C₆₁ ⋊ Dic₅
Hall	C₅×C₆₁ ⋊ Dic₉
Sylow	C₆₁ ⋊ Dic₄₅
Hall	C₅×C₉ ⋊ Dic₆₁
Sylow	C₉ ⋊ Dic₃₀₅
Sylow	C₅ ⋊ Dic₅₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₇₄₅
commutator, hypocenter	C₅×C₉×C₆₁	C₄
Frattini	C₂×C₃	D₉₁₅
Fitting, cosocle	C₂×C₅×C₉×C₆₁	C₂
socle, prime radical	C₂×C₃×C₅×C₆₁	D₃
LCS	G ▷ C₅×C₉×C₆₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 10980

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,3,5,61,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^60,f^a=f^2

Conjugacy classes & character table

2748 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₄₉₀, Dic₈₂₃₅, Dic₁₃₇₂₅
maximal normal subgroup of	Dic₅₄₉₀
maximal quotient of	Dic₈₂₃₅, Dic₁₃₇₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10980, 18);

Dic2745 #24926668