Dic₆₀₉₅ #37567170

order	24380
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₉₅
unambiguous semidirect product	C₂₃⋊Dic₂₆₅

Splittings

derived, Hall	C₅×C₂₃×C₅₃ ⋊ C₄
Hall	C₂₃×C₅₃ ⋊ Dic₅
Hall	C₅×C₅₃ ⋊ Dic₂₃
Hall	C₅×C₂₃ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₁₅
Sylow	C₂₃ ⋊ Dic₂₆₅
Sylow	C₅ ⋊ Dic₁₂₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₀₉₅
commutator, hypocenter	C₅×C₂₃×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₃×C₅₃	C₂
LCS	G ▷ C₅×C₂₃×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 24380

A minimal faithful permutation representation — 2 generators on 85 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);(1,3,2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,5,23,53,2|a^2=e,b^a=b^4,c^a=c^22,d^a=d^52

Conjugacy classes & character table

6098 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₁₉₀
maximal normal subgroup of	Dic₁₂₁₉₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24380, 8);

Dic6095 #37567170