Dic₃₀₅₅ #27132012

order	12220
automorphism	Aut(G) order 13490880
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₀₅₅
unambiguous semidirect product	C₄₇⋊Dic₆₅

Splittings

derived, Hall	C₅×C₁₃×C₄₇ ⋊ C₄
Hall	C₁₃×C₄₇ ⋊ Dic₅
Hall	C₅×C₄₇ ⋊ Dic₁₃
Hall	C₅×C₁₃ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₆₅
Sylow	C₁₃ ⋊ Dic₂₃₅
Sylow	C₅ ⋊ Dic₆₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₀₅₅
commutator, hypocenter	C₅×C₁₃×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₄₇	C₂
LCS	G ▷ C₅×C₁₃×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 12220

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

Polycyclic presentation:

2,5,13,47,2|a^2=e,b^a=b^4,c^a=c^12,d^a=d^46

Conjugacy classes & character table

3058 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₃₀₅₅, Dic₉₁₆₅, C₄₇⋊(Dic₆₅⋊C₃), …
maximal normal subgroup of	C₃×Dic₃₀₅₅, C₄₇⋊(Dic₆₅⋊C₃), Dic₆₁₁₀
maximal quotient of	C₃×Dic₃₀₅₅, Dic₉₁₆₅, Dic₁₅₂₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12220, 8);

Dic3055 #27132012