Dic₂₇₀₉ #24925438

order	10836
automorphism	Aut(G) order 8192016
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₀₉
unambiguous semidirect product	C₇⋊Dic₃₈₇
unambiguous semidirect product	C₉⋊Dic₃₀₁
unambiguous semidirect product	C₄₃⋊Dic₆₃

Splittings

derived, Hall	C₇×C₉×C₄₃ ⋊ C₄
Hall	C₉×C₄₃ ⋊ Dic₇
Hall	C₇×C₄₃ ⋊ Dic₉
Hall	C₇×C₉ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₆₃
Sylow	C₉ ⋊ Dic₃₀₁
Sylow	C₇ ⋊ Dic₃₈₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₇₀₉
commutator, hypocenter	C₇×C₉×C₄₃	C₄
Frattini	C₂×C₃	D₉₀₃
Fitting, cosocle	C₂×C₇×C₉×C₄₃	C₂
socle, prime radical	C₂×C₃×C₇×C₄₃	D₃
LCS	G ▷ C₇×C₉×C₄₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 10836

A minimal faithful permutation representation — 2 generators on 63 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,20)(16,19)(17,18)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Polycyclic presentation:

2,3,7,43,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^6,d^a=d^42,f^a=f^2

Conjugacy classes & character table

2712 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₄₁₈, Dic₈₁₂₇, Dic₁₃₅₄₅
maximal normal subgroup of	Dic₅₄₁₈
maximal quotient of	Dic₈₁₂₇, Dic₁₃₅₄₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10836, 30);

Dic2709 #24925438