Dic₅₇₈₅ #37528947

order	23140
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₇₈₅

Splittings

derived, Hall	C₅×C₁₃×C₈₉ ⋊ C₄
Hall	C₁₃×C₈₉ ⋊ Dic₅
Hall	C₅×C₈₉ ⋊ Dic₁₃
Sylow	C₈₉ ⋊ Dic₆₅
Hall	C₅×C₁₃ ⋊ Dic₈₉
Sylow	C₁₃ ⋊ Dic₄₄₅
Sylow	C₅ ⋊ Dic₁₁₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₈₅
commutator, hypocenter	C₅×C₁₃×C₈₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₈₉	C₂
LCS	G ▷ C₅×C₁₃×C₈₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 23140

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2,5,13,89,2|a^2=e,b^a=b^4,c^a=c^12,d^a=d^88

Conjugacy classes & character table

5788 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₅₇₀
maximal normal subgroup of	Dic₁₁₅₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23140, 8);

Dic5785 #37528947