Dic₂₇₁₇ #24925875

order	10868
automorphism	Aut(G) order 11737440
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₁₇
unambiguous semidirect product	C₁₁⋊Dic₂₄₇
unambiguous semidirect product	C₁₉⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₁₉ ⋊ C₄
Hall	C₁₃×C₁₉ ⋊ Dic₁₁
Hall	C₁₁×C₁₉ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₂₀₉
Sylow	C₁₁ ⋊ Dic₂₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₁₇
commutator, hypocenter	C₁₁×C₁₃×C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₁₉	C₂
LCS	G ▷ C₁₁×C₁₃×C₁₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 10868

A minimal faithful permutation representation — 2 generators on 47 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Polycyclic presentation:

2,11,13,19,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^18

Conjugacy classes & character table

2720 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₁₇, Dic₈₁₅₁, C₁₁⋊((C₁₉⋊C₃)⋊Dic₁₃), …
maximal normal subgroup of	C₃×Dic₂₇₁₇, C₁₁⋊((C₁₉⋊C₃)⋊Dic₁₃), C₁₁⋊#15688282, …
maximal quotient of	C₃×Dic₂₇₁₇, Dic₈₁₅₁, Dic₁₃₅₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10868, 8);

Dic2717 #24925875