Dic₇₉₅₅ #44541303

order	31820
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₉₅₅
unambiguous semidirect product	C₄₃⋊Dic₁₈₅

Splittings

derived, Hall	C₅×C₃₇×C₄₃ ⋊ C₄
Hall	C₃₇×C₄₃ ⋊ Dic₅
Hall	C₅×C₄₃ ⋊ Dic₃₇
Hall	C₅×C₃₇ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₈₅
Sylow	C₃₇ ⋊ Dic₂₁₅
Sylow	C₅ ⋊ Dic₁₅₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₉₅₅
commutator, hypocenter	C₅×C₃₇×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₃₇×C₄₃	C₂
LCS	G ▷ C₅×C₃₇×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 31820

A minimal faithful permutation representation — 2 generators on 89 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);(1,3,2,4)(6,9)(7,8)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2,5,37,43,2|a^2=e,b^a=b^4,c^a=c^36,d^a=d^42

Conjugacy classes & character table

7958 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₉₁₀
maximal normal subgroup of	Dic₁₅₉₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31820, 8);

Dic7955 #44541303