Dic₈₀₂₉ #44545793

order	32116
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₂₉
unambiguous semidirect product	C₇⋊Dic₁₁₄₇
unambiguous semidirect product	C₃₁⋊Dic₂₅₉

Splittings

derived, Hall	C₇×C₃₁×C₃₇ ⋊ C₄
Hall	C₃₁×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₃₁
Hall	C₇×C₃₁ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₂₁₇
Sylow	C₃₁ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₁₁₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₂₉
commutator, hypocenter	C₇×C₃₁×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₃₁×C₃₇	C₂
LCS	G ▷ C₇×C₃₁×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 32116

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)(6,11)(7,10)(8,9)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2,7,31,37,2|a^2=e,b^a=b^6,c^a=c^30,d^a=d^36

Conjugacy classes & character table

8032 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₀₅₈
maximal normal subgroup of	Dic₁₆₀₅₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32116, 8);

Dic8029 #44545793