Dic₄₈₀₇ #34063642

order	19228
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₈₀₇
unambiguous semidirect product	C₁₁⋊Dic₄₃₇
unambiguous semidirect product	C₁₉⋊Dic₂₅₃
unambiguous semidirect product	C₂₃⋊Dic₂₀₉

Splittings

derived, Hall	C₁₁×C₁₉×C₂₃ ⋊ C₄
Hall	C₁₉×C₂₃ ⋊ Dic₁₁
Hall	C₁₁×C₂₃ ⋊ Dic₁₉
Hall	C₁₁×C₁₉ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₂₀₉
Sylow	C₁₉ ⋊ Dic₂₅₃
Sylow	C₁₁ ⋊ Dic₄₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₈₀₇
commutator, hypocenter	C₁₁×C₁₉×C₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₉×C₂₃	C₂
LCS	G ▷ C₁₁×C₁₉×C₂₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 19228

A minimal faithful permutation representation — 2 generators on 57 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,11,19,23,2|a^2=e,b^a=b^10,c^a=c^18,d^a=d^22

Conjugacy classes & character table

4810 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₆₁₄, Dic₁₄₄₂₁
maximal normal subgroup of	Dic₉₆₁₄
maximal quotient of	Dic₁₄₄₂₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19228, 9);

Dic4807 #34063642