Dic₇₈₈₅ #44538692

order	31540
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₈₅
unambiguous semidirect product	C₁₉⋊Dic₄₁₅
unambiguous semidirect product	C₈₃⋊Dic₉₅

Splittings

derived, Hall	C₅×C₁₉×C₈₃ ⋊ C₄
Hall	C₁₉×C₈₃ ⋊ Dic₅
Hall	C₅×C₈₃ ⋊ Dic₁₉
Hall	C₅×C₁₉ ⋊ Dic₈₃
Sylow	C₈₃ ⋊ Dic₉₅
Sylow	C₁₉ ⋊ Dic₄₁₅
Sylow	C₅ ⋊ Dic₁₅₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₈₅
commutator, hypocenter	C₅×C₁₉×C₈₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₉×C₈₃	C₂
LCS	G ▷ C₅×C₁₉×C₈₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 31540

A minimal faithful permutation representation — 2 generators on 111 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);(1,3,2,4)(6,9)(7,8)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2,5,19,83,2|a^2=e,b^a=b^4,c^a=c^18,d^a=d^82

Conjugacy classes & character table

7888 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₇₇₀
maximal normal subgroup of	Dic₁₅₇₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31540, 8);

Dic7885 #44538692