Dic₅₈₈₅ #37555320

order	23540
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₈₈₅
unambiguous semidirect product	C₁₁⋊Dic₅₃₅
unambiguous semidirect product	C₁₀₇⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₁₀₇ ⋊ C₄
Hall	C₁₁×C₁₀₇ ⋊ Dic₅
Hall	C₅×C₁₀₇ ⋊ Dic₁₁
Sylow	C₁₀₇ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₁₀₇
Sylow	C₁₁ ⋊ Dic₅₃₅
Sylow	C₅ ⋊ Dic₁₁₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₈₈₅
commutator, hypocenter	C₅×C₁₁×C₁₀₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₁₀₇	C₂
LCS	G ▷ C₅×C₁₁×C₁₀₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 23540

A minimal faithful permutation representation — 2 generators on 127 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,127)(23,126)(24,125)(25,124)(26,123)(27,122)(28,121)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2,5,11,107,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^106

Conjugacy classes & character table

5888 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₇₇₀
maximal normal subgroup of	Dic₁₁₇₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23540, 9);

Dic5885 #37555320