Dic₇₅₇₉ #44476337

order	30316
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₅₇₉
unambiguous semidirect product	C₁₁⋊Dic₆₈₉

Splittings

derived, Hall	C₁₁×C₁₃×C₅₃ ⋊ C₄
Hall	C₁₃×C₅₃ ⋊ Dic₁₁
Hall	C₁₁×C₅₃ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₅₈₃
Sylow	C₁₁ ⋊ Dic₆₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₅₇₉
commutator, hypocenter	C₁₁×C₁₃×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₅₃	C₂
LCS	G ▷ C₁₁×C₁₃×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 30316

A minimal faithful permutation representation — 2 generators on 81 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,11,13,53,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^52

Conjugacy classes & character table

7582 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₁₅₈
maximal normal subgroup of	Dic₁₅₁₅₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30316, 9);

Dic7579 #44476337