Dic₃₉₃₃ #30628547

order	15732
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₉₃₃
unambiguous semidirect product	C₉⋊Dic₄₃₇
unambiguous semidirect product	C₁₉⋊Dic₂₀₇
unambiguous semidirect product	C₂₃⋊Dic₁₇₁

Splittings

derived, Hall	C₉×C₁₉×C₂₃ ⋊ C₄
Hall	C₁₉×C₂₃ ⋊ Dic₉
Hall	C₉×C₂₃ ⋊ Dic₁₉
Hall	C₉×C₁₉ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₁₇₁
Sylow	C₁₉ ⋊ Dic₂₀₇
Sylow	C₉ ⋊ Dic₄₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₉₃₃
commutator, hypocenter	C₉×C₁₉×C₂₃	C₄
Frattini	C₂×C₃	D₁₃₁₁
Fitting, cosocle	C₂×C₉×C₁₉×C₂₃	C₂
socle, prime radical	C₂×C₃×C₁₉×C₂₃	D₃
LCS	G ▷ C₉×C₁₉×C₂₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 15732

A minimal faithful permutation representation — 2 generators on 55 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2,3,19,23,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^18,d^a=d^22,f^a=f^2

Conjugacy classes & character table

3936 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₈₆₆, Dic₁₁₇₉₉
maximal normal subgroup of	Dic₇₈₆₆
maximal quotient of	Dic₁₁₇₉₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15732, 9);

Dic3933 #30628547