Dic₄₅₉₉ #32913694

order	18396
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₉₉
unambiguous semidirect product	C₇⋊Dic₆₅₇
unambiguous semidirect product	C₉⋊Dic₅₁₁

Splittings

derived, Hall	C₇×C₉×C₇₃ ⋊ C₄
Hall	C₉×C₇₃ ⋊ Dic₇
Hall	C₇×C₇₃ ⋊ Dic₉
Sylow	C₇₃ ⋊ Dic₆₃
Hall	C₇×C₉ ⋊ Dic₇₃
Sylow	C₉ ⋊ Dic₅₁₁
Sylow	C₇ ⋊ Dic₆₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₅₉₉
commutator, hypocenter	C₇×C₉×C₇₃	C₄
Frattini	C₂×C₃	D₁₅₃₃
Fitting, cosocle	C₂×C₇×C₉×C₇₃	C₂
socle, prime radical	C₂×C₃×C₇×C₇₃	D₃
LCS	G ▷ C₇×C₉×C₇₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 18396

A minimal faithful permutation representation — 2 generators on 93 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,20)(16,19)(17,18)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(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:

2,3,7,73,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^6,d^a=d^72,f^a=f^2

Conjugacy classes & character table

4602 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₁₉₈, Dic₁₃₇₉₇
maximal normal subgroup of	Dic₉₁₉₈
maximal quotient of	Dic₁₃₇₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18396, 12);

Dic4599 #32913694