Dic₅₂₄₇ #35230549

order	20988
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₂₄₇
unambiguous semidirect product	C₉⋊Dic₅₈₃
unambiguous semidirect product	C₁₁⋊Dic₄₇₇

Splittings

derived, Hall	C₉×C₁₁×C₅₃ ⋊ C₄
Hall	C₁₁×C₅₃ ⋊ Dic₉
Hall	C₉×C₅₃ ⋊ Dic₁₁
Hall	C₉×C₁₁ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₉₉
Sylow	C₁₁ ⋊ Dic₄₇₇
Sylow	C₉ ⋊ Dic₅₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₂₄₇
commutator, hypocenter	C₉×C₁₁×C₅₃	C₄
Frattini	C₂×C₃	D₁₇₄₉
Fitting, cosocle	C₂×C₉×C₁₁×C₅₃	C₂
socle, prime radical	C₂×C₃×C₁₁×C₅₃	D₃
LCS	G ▷ C₉×C₁₁×C₅₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 20988

A minimal faithful permutation representation — 2 generators on 77 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,24)(16,23)(17,22)(18,21)(19,20)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,3,11,53,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^10,d^a=d^52,f^a=f^2

Conjugacy classes & character table

5250 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₄₉₄, Dic₁₅₇₄₁
maximal normal subgroup of	Dic₁₀₄₉₄
maximal quotient of	Dic₁₅₇₄₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20988, 8);

Dic5247 #35230549