Dic₈₀₃₇ #44565249

order	32148
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₃₇
unambiguous semidirect product	C₉⋊Dic₈₉₃
unambiguous semidirect product	C₁₉⋊Dic₄₂₃
unambiguous semidirect product	C₄₇⋊Dic₁₇₁

Splittings

derived, Hall	C₉×C₁₉×C₄₇ ⋊ C₄
Hall	C₁₉×C₄₇ ⋊ Dic₉
Hall	C₉×C₄₇ ⋊ Dic₁₉
Hall	C₉×C₁₉ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₇₁
Sylow	C₁₉ ⋊ Dic₄₂₃
Sylow	C₉ ⋊ Dic₈₉₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₀₃₇
commutator, hypocenter	C₉×C₁₉×C₄₇	C₄
Frattini	C₂×C₃	D₂₆₇₉
Fitting, cosocle	C₂×C₉×C₁₉×C₄₇	C₂
socle, prime radical	C₂×C₃×C₁₉×C₄₇	D₃
LCS	G ▷ C₉×C₁₉×C₄₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 32148

A minimal faithful permutation representation — 2 generators on 79 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(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)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,3,19,47,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^18,d^a=d^46,f^a=f^2

Conjugacy classes & character table

8040 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₀₇₄
maximal normal subgroup of	Dic₁₆₀₇₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32148, 9);

Dic8037 #44565249