Dic₅₀₂₅ #34117153

order	20100
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₀₂₅
unambiguous semidirect product	C₃⋊Dic₁₆₇₅
unambiguous semidirect product	C₆₇⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₆₇ ⋊ C₄
Hall	C₂₅×C₆₇ ⋊ Dic₃
Hall	C₃×C₆₇ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₂₀₁
Sylow	C₃ ⋊ Dic₁₆₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₀₂₅
commutator, hypocenter	C₃×C₂₅×C₆₇	C₄
Frattini	C₂×C₅	D₁₀₀₅
Fitting, cosocle	C₂×C₃×C₂₅×C₆₇	C₂
socle, prime radical	C₂×C₃×C₅×C₆₇	D₅
LCS	G ▷ C₃×C₂₅×C₆₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 20100

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

Polycyclic presentation:

2,3,5,67,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^66,f^a=f^4

Conjugacy classes & character table

5028 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₅₀, Dic₁₅₀₇₅
maximal normal subgroup of	Dic₁₀₀₅₀
maximal quotient of	Dic₁₅₀₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 9);

Dic5025 #34117153