Dic₂₉₀₇ #26042564

order	11628
automorphism	Aut(G) order 10046592
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₉₀₇
unambiguous semidirect product	C₉⋊Dic₃₂₃
unambiguous semidirect product	C₁₉⋊Dic₁₅₃

Splittings

derived, Hall	C₉×C₁₇×C₁₉ ⋊ C₄
Hall	C₁₇×C₁₉ ⋊ Dic₉
Hall	C₉×C₁₉ ⋊ Dic₁₇
Hall	C₉×C₁₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₁₅₃
Sylow	C₁₇ ⋊ Dic₁₇₁
Sylow	C₉ ⋊ Dic₃₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₉₀₇
commutator, hypocenter	C₉×C₁₇×C₁₉	C₄
Frattini	C₂×C₃	D₉₆₉
Fitting, cosocle	C₂×C₉×C₁₇×C₁₉	C₂
socle, prime radical	C₂×C₃×C₁₇×C₁₉	D₃
LCS	G ▷ C₉×C₁₇×C₁₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 11628

A minimal faithful permutation representation — 2 generators on 49 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2,3,17,19,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^16,d^a=d^18,f^a=f^2

Conjugacy classes & character table

2910 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₈₁₄, Dic₈₇₂₁, Dic₁₄₅₃₅
maximal normal subgroup of	Dic₅₈₁₄
maximal quotient of	Dic₈₇₂₁, Dic₁₄₅₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11628, 12);

Dic2907 #26042564