Dic₃₁₄₅ #27160356

order	12580
automorphism	Aut(G) order 14492160
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₄₅

Splittings

derived, Hall	C₅×C₁₇×C₃₇ ⋊ C₄
Hall	C₁₇×C₃₇ ⋊ Dic₅
Hall	C₅×C₃₇ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₁₈₅
Sylow	C₅ ⋊ Dic₆₂₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₁₄₅
commutator, hypocenter	C₅×C₁₇×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₃₇	C₂
LCS	G ▷ C₅×C₁₇×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 12580

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

Polycyclic presentation:

2,5,17,37,2|a^2=e,b^a=b^4,c^a=c^16,d^a=d^36

Conjugacy classes & character table

3148 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₃₁₄₅, Dic₉₄₃₅, Dic₃₁₄₅⋊C₃, …
maximal normal subgroup of	C₃×Dic₃₁₄₅, Dic₃₁₄₅⋊C₃, Dic₆₂₉₀
maximal quotient of	C₃×Dic₃₁₄₅, Dic₉₄₃₅, Dic₁₅₇₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12580, 8);

Dic3145 #27160356