Dic₆₇₄₅ #41008682

order	26980
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₇₄₅
unambiguous semidirect product	C₁₉⋊Dic₃₅₅
unambiguous semidirect product	C₇₁⋊Dic₉₅

Splittings

derived, Hall	C₅×C₁₉×C₇₁ ⋊ C₄
Hall	C₁₉×C₇₁ ⋊ Dic₅
Hall	C₅×C₇₁ ⋊ Dic₁₉
Hall	C₅×C₁₉ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₉₅
Sylow	C₁₉ ⋊ Dic₃₅₅
Sylow	C₅ ⋊ Dic₁₃₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₇₄₅
commutator, hypocenter	C₅×C₁₉×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₉×C₇₁	C₂
LCS	G ▷ C₅×C₁₉×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 26980

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)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2,5,19,71,2|a^2=e,b^a=b^4,c^a=c^18,d^a=d^70

Conjugacy classes & character table

6748 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₄₉₀
maximal normal subgroup of	Dic₁₃₄₉₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26980, 9);

Dic6745 #41008682