Dic₄₇₄₃ #34039006

order	18972
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₇₄₃
unambiguous semidirect product	C₉⋊Dic₅₂₇
unambiguous semidirect product	C₃₁⋊Dic₁₅₃

Splittings

derived, Hall	C₉×C₁₇×C₃₁ ⋊ C₄
Hall	C₁₇×C₃₁ ⋊ Dic₉
Hall	C₉×C₃₁ ⋊ Dic₁₇
Hall	C₉×C₁₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₅₃
Sylow	C₁₇ ⋊ Dic₂₇₉
Sylow	C₉ ⋊ Dic₅₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₇₄₃
commutator, hypocenter	C₉×C₁₇×C₃₁	C₄
Frattini	C₂×C₃	D₁₅₈₁
Fitting, cosocle	C₂×C₉×C₁₇×C₃₁	C₂
socle, prime radical	C₂×C₃×C₁₇×C₃₁	D₃
LCS	G ▷ C₉×C₁₇×C₃₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 18972

A minimal faithful permutation representation — 2 generators on 61 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(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:

2,3,17,31,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^16,d^a=d^30,f^a=f^2

Conjugacy classes & character table

4746 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₄₈₆, Dic₁₄₂₂₉
maximal normal subgroup of	Dic₉₄₈₆
maximal quotient of	Dic₁₄₂₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18972, 9);

Dic4743 #34039006