Dic₅₀₃₁ #34117409

order	20124
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₀₃₁
unambiguous semidirect product	C₉⋊Dic₅₅₉
unambiguous semidirect product	C₄₃⋊Dic₁₁₇

Splittings

derived, Hall	C₉×C₁₃×C₄₃ ⋊ C₄
Hall	C₁₃×C₄₃ ⋊ Dic₉
Hall	C₉×C₄₃ ⋊ Dic₁₃
Hall	C₉×C₁₃ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₁₇
Sylow	C₁₃ ⋊ Dic₃₈₇
Sylow	C₉ ⋊ Dic₅₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₀₃₁
commutator, hypocenter	C₉×C₁₃×C₄₃	C₄
Frattini	C₂×C₃	D₁₆₇₇
Fitting, cosocle	C₂×C₉×C₁₃×C₄₃	C₂
socle, prime radical	C₂×C₃×C₁₃×C₄₃	D₃
LCS	G ▷ C₉×C₁₃×C₄₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 20124

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)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,3,13,43,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^12,d^a=d^42,f^a=f^2

Conjugacy classes & character table

5034 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₆₂, Dic₁₅₀₉₃
maximal normal subgroup of	Dic₁₀₀₆₂
maximal quotient of	Dic₁₅₀₉₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 12);

Dic5031 #34117409