Dic₄₅₃₉ #31827754

order	18156
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₃₉
unambiguous semidirect product	C₃⋊Dic₁₅₁₃

Splittings

derived, Hall	C₃×C₁₇×C₈₉ ⋊ C₄
Hall	C₁₇×C₈₉ ⋊ Dic₃
Hall	C₃×C₈₉ ⋊ Dic₁₇
Sylow	C₈₉ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₈₉
Sylow	C₁₇ ⋊ Dic₂₆₇
Sylow	C₃ ⋊ Dic₁₅₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₅₃₉
commutator, hypocenter	C₃×C₁₇×C₈₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₈₉	C₂
LCS	G ▷ C₃×C₁₇×C₈₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 18156

A minimal faithful permutation representation — 2 generators on 113 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3,2,4)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2,3,17,89,2|a^2=e,b^a=b^2,c^a=c^16,d^a=d^88

Conjugacy classes & character table

4542 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₇₈, Dic₁₃₆₁₇
maximal normal subgroup of	Dic₉₀₇₈
maximal quotient of	Dic₁₃₆₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18156, 8);

Dic4539 #31827754