Dic₇₄₅₅ #44447133

order	29820
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₄₅₅
unambiguous semidirect product	C₃⋊Dic₂₄₈₅
unambiguous semidirect product	C₇⋊Dic₁₀₆₅
unambiguous semidirect product	C₇₁⋊Dic₁₀₅

Splittings

derived, Hall	C₃×C₅×C₇×C₇₁ ⋊ C₄
Hall	C₅×C₇×C₇₁ ⋊ Dic₃
Hall	C₃×C₇×C₇₁ ⋊ Dic₅
Hall	C₃×C₅×C₇₁ ⋊ Dic₇
Hall	C₇×C₇₁ ⋊ Dic₁₅
Hall	C₅×C₇₁ ⋊ Dic₂₁
Hall	C₃×C₇₁ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₂₁₃
Hall	C₃×C₇ ⋊ Dic₃₅₅
Hall	C₃×C₅ ⋊ Dic₄₉₇
Sylow	C₇ ⋊ Dic₁₀₆₅
Sylow	C₅ ⋊ Dic₁₄₉₁
Sylow	C₃ ⋊ Dic₂₄₈₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₄₅₅
commutator, hypocenter	C₃×C₅×C₇×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₇₁	C₂
LCS	G ▷ C₃×C₅×C₇×C₇₁
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 29820

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

Polycyclic presentation:

2,3,5,7,71,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^70

Conjugacy classes & character table

7458 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₉₁₀
maximal normal subgroup of	Dic₁₄₉₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 21);

Dic7455 #44447133