Dic₅₄₉₉ #36347168

order	21996
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₄₉₉
unambiguous semidirect product	C₉⋊Dic₆₁₁
unambiguous semidirect product	C₄₇⋊Dic₁₁₇

Splittings

derived, Hall	C₉×C₁₃×C₄₇ ⋊ C₄
Hall	C₁₃×C₄₇ ⋊ Dic₉
Hall	C₉×C₄₇ ⋊ Dic₁₃
Hall	C₉×C₁₃ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₁₇
Sylow	C₁₃ ⋊ Dic₄₂₃
Sylow	C₉ ⋊ Dic₆₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₄₉₉
commutator, hypocenter	C₉×C₁₃×C₄₇	C₄
Frattini	C₂×C₃	D₁₈₃₃
Fitting, cosocle	C₂×C₉×C₁₃×C₄₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₄₇	D₃
LCS	G ▷ C₉×C₁₃×C₄₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 21996

A minimal faithful permutation representation — 2 generators on 73 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,3,13,47,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^12,d^a=d^46,f^a=f^2

Conjugacy classes & character table

5502 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₉₉₈
maximal normal subgroup of	Dic₁₀₉₉₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21996, 9);

Dic5499 #36347168