Dic₄₆₉₇ #34037693

order	18788
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₆₉₇
unambiguous semidirect product	C₇⋊Dic₆₇₁
unambiguous semidirect product	C₁₁⋊Dic₄₂₇

Splittings

derived, Hall	C₇×C₁₁×C₆₁ ⋊ C₄
Hall	C₁₁×C₆₁ ⋊ Dic₇
Hall	C₇×C₆₁ ⋊ Dic₁₁
Hall	C₇×C₁₁ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₇₇
Sylow	C₁₁ ⋊ Dic₄₂₇
Sylow	C₇ ⋊ Dic₆₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₆₉₇
commutator, hypocenter	C₇×C₁₁×C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₆₁	C₂
LCS	G ▷ C₇×C₁₁×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 18788

A minimal faithful permutation representation — 2 generators on 83 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,7,11,61,2|a^2=e,b^a=b^6,c^a=c^10,d^a=d^60

Conjugacy classes & character table

4700 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₃₉₄, Dic₁₄₀₉₁
maximal normal subgroup of	Dic₉₃₉₄
maximal quotient of	Dic₁₄₀₉₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18788, 8);

Dic4697 #34037693