Dic₉₂₄₃ #49149366

order	36972
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₂₄₃
unambiguous semidirect product	C₉⋊Dic₁₀₂₇
unambiguous semidirect product	C₇₉⋊Dic₁₁₇

Splittings

derived, Hall	C₉×C₁₃×C₇₉ ⋊ C₄
Hall	C₁₃×C₇₉ ⋊ Dic₉
Hall	C₉×C₇₉ ⋊ Dic₁₃
Hall	C₉×C₁₃ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₁₇
Sylow	C₁₃ ⋊ Dic₇₁₁
Sylow	C₉ ⋊ Dic₁₀₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₉₂₄₃
commutator, hypocenter	C₉×C₁₃×C₇₉	C₄
Frattini	C₂×C₃	D₃₀₈₁
Fitting, cosocle	C₂×C₉×C₁₃×C₇₉	C₂
socle, prime radical	C₂×C₃×C₁₃×C₇₉	D₃
LCS	G ▷ C₉×C₁₃×C₇₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 36972

A minimal faithful permutation representation — 2 generators on 105 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2,3,13,79,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^12,d^a=d^78,f^a=f^2

Conjugacy classes & character table

9246 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36972, 14);

Dic9243 #49149366