Dic₃₄₇₇ #28358045

order	13908
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₄₇₇
unambiguous semidirect product	C₃⋊Dic₁₁₅₉
unambiguous semidirect product	C₁₉⋊Dic₁₈₃

Splittings

derived, Hall	C₃×C₁₉×C₆₁ ⋊ C₄
Hall	C₁₉×C₆₁ ⋊ Dic₃
Hall	C₃×C₆₁ ⋊ Dic₁₉
Sylow	C₆₁ ⋊ Dic₅₇
Hall	C₃×C₁₉ ⋊ Dic₆₁
Sylow	C₁₉ ⋊ Dic₁₈₃
Sylow	C₃ ⋊ Dic₁₁₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₄₇₇
commutator, hypocenter	C₃×C₁₉×C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₆₁	C₂
LCS	G ▷ C₃×C₁₉×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 13908

A minimal faithful permutation representation — 2 generators on 87 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);(1,3,2,4)(6,7)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,3,19,61,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^60

Conjugacy classes & character table

3480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₉₅₄
maximal normal subgroup of	Dic₆₉₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13908, 12);

Dic3477 #28358045