Dic₄₁₄₇ #30658689

order	16588
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₁₄₇
unambiguous semidirect product	C₁₁⋊Dic₃₇₇

Splittings

derived, Hall	C₁₁×C₁₃×C₂₉ ⋊ C₄
Hall	C₁₃×C₂₉ ⋊ Dic₁₁
Hall	C₁₁×C₂₉ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₃₁₉
Sylow	C₁₁ ⋊ Dic₃₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₁₄₇
commutator, hypocenter	C₁₁×C₁₃×C₂₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₂₉	C₂
LCS	G ▷ C₁₁×C₁₃×C₂₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 16588

A minimal faithful permutation representation — 2 generators on 57 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2,11,13,29,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^28

Conjugacy classes & character table

4150 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₂₉₄
maximal normal subgroup of	Dic₈₂₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16588, 8);

Dic4147 #30658689