Dic₂₇₀₃ #24925199

order	10812
automorphism	Aut(G) order 8995584
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₀₃
unambiguous semidirect product	C₃⋊Dic₉₀₁

Splittings

derived, Hall	C₃×C₁₇×C₅₃ ⋊ C₄
Hall	C₁₇×C₅₃ ⋊ Dic₃
Hall	C₃×C₅₃ ⋊ Dic₁₇
Sylow	C₅₃ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₅₃
Sylow	C₁₇ ⋊ Dic₁₅₉
Sylow	C₃ ⋊ Dic₉₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₀₃
commutator, hypocenter	C₃×C₁₇×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₅₃	C₂
LCS	G ▷ C₃×C₁₇×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 10812

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)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(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:

2,3,17,53,2|a^2=e,b^a=b^2,c^a=c^16,d^a=d^52

Conjugacy classes & character table

2706 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₁₀₉, C₃⋊Dic₂₇₀₃, C₃×Dic₂₇₀₃, …
maximal normal subgroup of	C₃×Dic₂₇₀₃, Dic₅₄₀₆
maximal quotient of	Dic₈₁₀₉, C₃⋊Dic₂₇₀₃, C₃×Dic₂₇₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10812, 8);

Dic2703 #24925199