Dic₅₇₁₉ #37504221

order	22876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₁₉
unambiguous semidirect product	C₇⋊Dic₈₁₇
unambiguous semidirect product	C₁₉⋊Dic₃₀₁
unambiguous semidirect product	C₄₃⋊Dic₁₃₃

Splittings

derived, Hall	C₇×C₁₉×C₄₃ ⋊ C₄
Hall	C₁₉×C₄₃ ⋊ Dic₇
Hall	C₇×C₄₃ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₃₀₁
Sylow	C₇ ⋊ Dic₈₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₁₉
commutator, hypocenter	C₇×C₁₉×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₄₃	C₂
LCS	G ▷ C₇×C₁₉×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 22876

A minimal faithful permutation representation — 2 generators on 73 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,7,19,43,2|a^2=e,b^a=b^6,c^a=c^18,d^a=d^42

Conjugacy classes & character table

5722 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₄₃₈
maximal normal subgroup of	Dic₁₁₄₃₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22876, 9);

Dic5719 #37504221