Dic₇₆₆₅ #44501257

order	30660
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₆₆₅
unambiguous semidirect product	C₃⋊Dic₂₅₅₅
unambiguous semidirect product	C₇⋊Dic₁₀₉₅

Splittings

derived, Hall	C₃×C₅×C₇×C₇₃ ⋊ C₄
Hall	C₅×C₇×C₇₃ ⋊ Dic₃
Hall	C₃×C₇×C₇₃ ⋊ Dic₅
Hall	C₃×C₅×C₇₃ ⋊ Dic₇
Hall	C₇×C₇₃ ⋊ Dic₁₅
Hall	C₅×C₇₃ ⋊ Dic₂₁
Hall	C₃×C₇₃ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₂₁₉
Hall	C₃×C₇ ⋊ Dic₃₆₅
Hall	C₃×C₅ ⋊ Dic₅₁₁
Sylow	C₇ ⋊ Dic₁₀₉₅
Sylow	C₅ ⋊ Dic₁₅₃₃
Sylow	C₃ ⋊ Dic₂₅₅₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₆₅
commutator, hypocenter	C₃×C₅×C₇×C₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₇₃	C₂
LCS	G ▷ C₃×C₅×C₇×C₇₃
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 30660

A minimal faithful permutation representation — 2 generators on 92 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92);(1,3,2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,3,5,7,73,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^72

Conjugacy classes & character table

7668 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₃₃₀
maximal normal subgroup of	Dic₁₅₃₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30660, 20);

Dic7665 #44501257