Dic₆₃₆₃ #38764115

order	25452
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₃₆₃
unambiguous semidirect product	C₇⋊Dic₉₀₉
unambiguous semidirect product	C₉⋊Dic₇₀₇

Splittings

derived, Hall	C₇×C₉×C₁₀₁ ⋊ C₄
Hall	C₉×C₁₀₁ ⋊ Dic₇
Hall	C₇×C₁₀₁ ⋊ Dic₉
Sylow	C₁₀₁ ⋊ Dic₆₃
Hall	C₇×C₉ ⋊ Dic₁₀₁
Sylow	C₉ ⋊ Dic₇₀₇
Sylow	C₇ ⋊ Dic₉₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₃₆₃
commutator, hypocenter	C₇×C₉×C₁₀₁	C₄
Frattini	C₂×C₃	D₂₁₂₁
Fitting, cosocle	C₂×C₇×C₉×C₁₀₁	C₂
socle, prime radical	C₂×C₃×C₇×C₁₀₁	D₃
LCS	G ▷ C₇×C₉×C₁₀₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 25452

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

Polycyclic presentation:

2,3,7,101,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^6,d^a=d^100,f^a=f^2

Conjugacy classes & character table

6366 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₇₂₆
maximal normal subgroup of	Dic₁₂₇₂₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25452, 9);

Dic6363 #38764115