Dic₇₈₄₃ #44535872

order	31372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₄₃
unambiguous semidirect product	C₁₁⋊Dic₇₁₃
unambiguous semidirect product	C₂₃⋊Dic₃₄₁
unambiguous semidirect product	C₃₁⋊Dic₂₅₃

Splittings

derived, Hall	C₁₁×C₂₃×C₃₁ ⋊ C₄
Hall	C₂₃×C₃₁ ⋊ Dic₁₁
Hall	C₁₁×C₃₁ ⋊ Dic₂₃
Hall	C₁₁×C₂₃ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₅₃
Sylow	C₂₃ ⋊ Dic₃₄₁
Sylow	C₁₁ ⋊ Dic₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₄₃
commutator, hypocenter	C₁₁×C₂₃×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₂₃×C₃₁	C₂
LCS	G ▷ C₁₁×C₂₃×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 31372

A minimal faithful permutation representation — 2 generators on 69 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,11,23,31,2|a^2=e,b^a=b^10,c^a=c^22,d^a=d^30

Conjugacy classes & character table

7846 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₆₈₆
maximal normal subgroup of	Dic₁₅₆₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31372, 9);

Dic7843 #44535872