Dic₆₂₀₅ #37613849

order	24820
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₂₀₅

Splittings

derived, Hall	C₅×C₁₇×C₇₃ ⋊ C₄
Hall	C₁₇×C₇₃ ⋊ Dic₅
Hall	C₅×C₇₃ ⋊ Dic₁₇
Hall	C₅×C₁₇ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₈₅
Sylow	C₁₇ ⋊ Dic₃₆₅
Sylow	C₅ ⋊ Dic₁₂₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₂₀₅
commutator, hypocenter	C₅×C₁₇×C₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₇₃	C₂
LCS	G ▷ C₅×C₁₇×C₇₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 24820

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

Polycyclic presentation:

2,5,17,73,2|a^2=e,b^a=b^4,c^a=c^16,d^a=d^72

Conjugacy classes & character table

6208 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₄₁₀
maximal normal subgroup of	Dic₁₂₄₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24820, 8);

Dic6205 #37613849