Dic₅₁₅₁ #35207248

order	20604
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₅₁
unambiguous semidirect product	C₃⋊Dic₁₇₁₇

Splittings

derived, Hall	C₃×C₁₇×C₁₀₁ ⋊ C₄
Hall	C₁₇×C₁₀₁ ⋊ Dic₃
Hall	C₃×C₁₀₁ ⋊ Dic₁₇
Sylow	C₁₀₁ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₁₀₁
Sylow	C₁₇ ⋊ Dic₃₀₃
Sylow	C₃ ⋊ Dic₁₇₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₅₁
commutator, hypocenter	C₃×C₁₇×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₁₀₁	C₂
LCS	G ▷ C₃×C₁₇×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 20604

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

Polycyclic presentation:

2,3,17,101,2|a^2=e,b^a=b^2,c^a=c^16,d^a=d^100

Conjugacy classes & character table

5154 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₀₂, Dic₁₅₄₅₃
maximal normal subgroup of	Dic₁₀₃₀₂
maximal quotient of	Dic₁₅₄₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20604, 8);

Dic5151 #35207248